Subgroups — C₂×C₆₇×#3276 #30740286

Order 17152.

2742 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#137×C₆₇	C₂	1	2	yes	maximal, complemented
2	C₂×D₄×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#547×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#3276	C₂	1	16	yes	maximal, direct product factor
5	C₂×#494×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#470×C₆₇	C₂	1	2	yes	maximal, complemented
7	C₂×#462×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#461×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#452×C₆₇	C₂	1	2	yes	maximal, complemented
10	C₂×#451×C₆₇	C₂	1	2	yes	maximal, complemented
11	C₂×C₄×SD₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#436×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×#41×C₆₇	C₂²	1	2	yes	—
14	C₂³×Q₈×C₆₇	C₂²	1	2	yes	—
15	C₂×#137×C₆₇	—	2	16	no	—
16	C₂×#137×C₆₇	C₂²	1	16	yes	complemented
17	C₂²×Q₁₆×C₆₇	C₂²	1	2	yes	—
18	C₂²×Q₁₆×C₆₇	C₂²	1	2	yes	—
19	C₂²×Q₁₆×C₆₇	C₂²	1	2	yes	—
20	C₂²×SD₁₆×C₆₇	C₂²	1	2	yes	—
21	C₂²×SD₁₆×C₆₇	C₂²	1	1	yes	characteristic
22	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
23	D₄×Q₈×C₆₇	—	2	4	no	—
24	D₄×Q₈×C₆₇	C₂²	1	8	yes	complemented
25	#547×C₆₇	—	2	4	no	—
26	#547×C₆₇	C₂²	1	8	yes	—
27	C₂×C₄⋊Q₈×C₆₇	—	2	1	no	—
28	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
30	C₂×#123×C₆₇	—	2	2	no	—
31	C₂×C₂²⋊Q₈×C₆₇	—	2	2	no	—
32	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
33	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
34	C₂×#121×C₆₇	—	2	1	no	—
35	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
36	C₂×C₄×D₄×C₆₇	—	2	1	no	—
37	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
38	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
39	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
40	#494×C₆₇	C₂²	1	8	yes	complemented
41	#470×C₆₇	C₂²	1	16	yes	complemented
42	#462×C₆₇	C₂²	1	8	yes	complemented
43	#461×C₆₇	C₂²	1	8	yes	complemented
44	#452×C₆₇	C₂²	1	16	yes	complemented
45	#451×C₆₇	C₂²	1	16	yes	complemented
46	C₄×SD₁₆×C₆₇	C₂²	1	8	yes	complemented
47	#436×C₆₇	C₂²	1	8	yes	complemented
48	C₂×#106×C₆₇	C₂²	1	1	yes	characteristic
49	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
50	C₂×#103×C₆₇	C₂²	1	2	yes	—
51	C₂×#103×C₆₇	C₂²	1	2	yes	—
52	C₂×#103×C₆₇	C₂²	1	1	yes	characteristic
53	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
54	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
55	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
56	C₂×#41×C₆₇	—	2	4	no	—
57	C₂×#41×C₆₇	—	2	8	no	—
58	C₂×#41×C₆₇	—	4	4	no	—
59	C₂×#41×C₆₇	C₂³	1	8	yes	—
60	C₂²×Q₈×C₆₇	—	2	2	no	—
61	C₂²×Q₈×C₆₇	—	2	4	no	—
62	C₂²×Q₈×C₆₇	—	2	8	no	—
63	C₂²×Q₈×C₆₇	—	4	4	no	—
64	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
65	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
66	C₂²×Q₈×C₆₇	C₂³	1	8	yes	—
67	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
68	C₂²×D₄×C₆₇	—	2	2	no	—
69	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
70	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
71	C₂³×C₄×C₆₇	—	2	2	no	—
72	C₂³×C₄×C₆₇	—	2	1	no	—
73	C₂³×C₄×C₆₇	—	2	2	no	—
74	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
75	#137×C₆₇	—	2	64	no	—
76	C₂×Q₁₆×C₆₇	—	2	8	no	—
77	C₂×Q₁₆×C₆₇	—	2	8	no	—
78	C₂×Q₁₆×C₆₇	—	4	4	no	—
79	C₂×Q₁₆×C₆₇	C₂³	1	8	yes	—
80	C₂×Q₁₆×C₆₇	C₂³	1	8	yes	—
81	C₂×Q₁₆×C₆₇	C₂³	1	8	yes	—
82	C₂×SD₁₆×C₆₇	—	2	8	no	—
83	C₂×SD₁₆×C₆₇	D₄	1	8	yes	complemented
84	C₂×SD₁₆×C₆₇	C₂³	1	8	yes	—
85	C₂×SD₁₆×C₆₇	C₂³	1	4	yes	—
86	C₂×M₁₆×C₆₇	—	2	8	no	—
87	C₂×M₁₆×C₆₇	C₂³	1	8	yes	—
88	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
89	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
90	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
91	C₄⋊Q₈×C₆₇	—	2	8	no	—
92	C₄⋊Q₈×C₆₇	—	2	2	no	—
93	C₄⋊Q₈×C₆₇	—	2	2	no	—
94	C₄⋊Q₈×C₆₇	C₂³	1	4	yes	—
95	C₄⋊Q₈×C₆₇	C₂³	1	4	yes	—
96	#123×C₆₇	—	2	16	no	—
97	C₂²⋊Q₈×C₆₇	—	2	4	no	—
98	C₂²⋊Q₈×C₆₇	—	2	4	no	—
99	C₂²⋊Q₈×C₆₇	—	2	16	no	—
100	C₂²⋊Q₈×C₆₇	C₂³	1	8	yes	—
101	C₂²⋊Q₈×C₆₇	C₂³	1	8	yes	—
102	#121×C₆₇	—	2	8	no	—
103	C₄×Q₈×C₆₇	—	2	2	no	—
104	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
105	C₄×D₄×C₆₇	—	2	8	no	—
106	C₄×D₄×C₆₇	—	2	2	no	—
107	C₄×D₄×C₆₇	D₄	1	4	yes	—
108	C₄×D₄×C₆₇	C₂³	1	4	yes	—
109	C₄×D₄×C₆₇	C₂³	1	4	yes	—
110	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
111	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
112	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
113	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
114	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
115	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
116	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
117	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
118	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
119	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
120	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
121	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
122	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
123	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
124	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
125	C₂×C₄²×C₆₇	—	2	1	no	—
126	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
127	#106×C₆₇	C₂³	1	4	yes	—
128	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
129	#103×C₆₇	C₂³	1	4	yes	—
130	#103×C₆₇	C₂³	1	8	yes	—
131	#103×C₆₇	C₂³	1	8	yes	—
132	#102×C₆₇	C₂³	1	4	yes	—
133	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	—
134	C₄×C₈×C₆₇	C₂³	1	4	yes	—
135	C₂⁴×C₆₇	—	2	1	no	—
136	C₂⁴×C₆₇	C₂×D₄	1	2	yes	—
137	#41×C₆₇	—	2	16	no	—
138	#41×C₆₇	—	4	8	no	—
139	#41×C₆₇	—	4	16	no	—
140	C₂×Q₈×C₆₇	—	2	1	no	—
141	C₂×Q₈×C₆₇	—	2	8	no	—
142	C₂×Q₈×C₆₇	—	2	2	no	—
143	C₂×Q₈×C₆₇	—	2	2	no	—
144	C₂×Q₈×C₆₇	—	2	2	no	—
145	C₂×Q₈×C₆₇	—	2	16	no	—
146	C₂×Q₈×C₆₇	—	2	4	no	—
147	C₂×Q₈×C₆₇	—	2	4	no	—
148	C₂×Q₈×C₆₇	—	2	4	no	—
149	C₂×Q₈×C₆₇	—	2	4	no	—
150	C₂×Q₈×C₆₇	—	4	8	no	—
151	C₂×Q₈×C₆₇	—	4	8	no	—
152	C₂×Q₈×C₆₇	—	4	16	no	—
153	C₂×Q₈×C₆₇	C₂×D₄	1	4	yes	—
154	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
155	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
156	C₂×Q₈×C₆₇	C₂⁴	1	2	yes	—
157	C₂×D₄×C₆₇	—	2	8	no	—
158	C₂×D₄×C₆₇	—	2	4	no	—
159	C₂×D₄×C₆₇	—	2	1	no	—
160	C₂×D₄×C₆₇	—	2	2	no	—
161	C₂×D₄×C₆₇	—	4	8	no	—
162	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
163	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
164	C₂×D₄×C₆₇	C₂⁴	1	2	yes	—
165	C₂²×C₄×C₆₇	—	2	1	no	—
166	C₂²×C₄×C₆₇	—	2	4	no	—
167	C₂²×C₄×C₆₇	—	2	4	no	—
168	C₂²×C₄×C₆₇	—	2	8	no	—
169	C₂²×C₄×C₆₇	—	2	1	no	—
170	C₂²×C₄×C₆₇	—	2	1	no	—
171	C₂²×C₄×C₆₇	—	2	2	no	—
172	C₂²×C₄×C₆₇	—	2	2	no	—
173	C₂²×C₄×C₆₇	—	2	4	no	—
174	C₂²×C₄×C₆₇	—	2	8	no	—
175	C₂²×C₄×C₆₇	—	4	8	no	—
176	C₂²×C₄×C₆₇	—	4	8	no	—
177	C₂²×C₄×C₆₇	—	4	4	no	—
178	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
179	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
180	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
181	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
182	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
183	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
184	Q₁₆×C₆₇	—	2	16	no	—
185	Q₁₆×C₆₇	—	2	16	no	—
186	Q₁₆×C₆₇	—	4	8	no	—
187	SD₁₆×C₆₇	—	2	16	no	—
188	SD₁₆×C₆₇	C₂×D₄	1	16	yes	complemented
189	M₁₆×C₆₇	—	2	16	no	—
190	C₂×C₈×C₆₇	—	2	4	no	—
191	C₂×C₈×C₆₇	—	2	2	no	—
192	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
193	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
194	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
195	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
196	C₄⋊C₄×C₆₇	—	2	4	no	—
197	C₄⋊C₄×C₆₇	—	2	8	no	—
198	C₄⋊C₄×C₆₇	—	2	4	no	—
199	C₄⋊C₄×C₆₇	—	2	4	no	—
200	C₄⋊C₄×C₆₇	—	2	4	no	—
201	C₄⋊C₄×C₆₇	—	2	8	no	—
202	C₄⋊C₄×C₆₇	—	2	4	no	—
203	C₄⋊C₄×C₆₇	—	2	1	no	—
204	C₄⋊C₄×C₆₇	—	2	1	no	—
205	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
206	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
207	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
208	C₂²⋊C₄×C₆₇	—	2	8	no	—
209	C₂²⋊C₄×C₆₇	—	2	8	no	—
210	C₂²⋊C₄×C₆₇	—	2	4	no	—
211	C₂²⋊C₄×C₆₇	C₂×D₄	1	8	yes	—
212	C₄²×C₆₇	—	2	4	no	—
213	C₄²×C₆₇	C₂×D₄	1	2	yes	—
214	C₄²×C₆₇	C₂⁴	1	2	yes	—
215	C₂³×C₆₇	—	2	4	no	—
216	C₂³×C₆₇	—	2	4	no	—
217	C₂³×C₆₇	—	2	4	no	—
218	C₂³×C₆₇	—	2	4	no	—
219	C₂³×C₆₇	—	2	2	no	—
220	C₂³×C₆₇	—	4	2	no	—
221	C₂³×C₆₇	—	4	2	no	—
222	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
223	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
224	Q₈×C₆₇	—	2	4	no	—
225	Q₈×C₆₇	—	2	2	no	—
226	Q₈×C₆₇	—	2	4	no	—
227	Q₈×C₆₇	—	2	4	no	—
228	Q₈×C₆₇	—	2	4	no	—
229	Q₈×C₆₇	—	2	4	no	—
230	Q₈×C₆₇	—	4	16	no	—
231	Q₈×C₆₇	—	4	4	no	—
232	Q₈×C₆₇	—	4	4	no	—
233	Q₈×C₆₇	C₂²×D₄	1	4	yes	—
234	D₄×C₆₇	—	2	4	no	—
235	D₄×C₆₇	—	2	8	no	—
236	D₄×C₆₇	—	2	2	no	—
237	D₄×C₆₇	—	4	16	no	—
238	D₄×C₆₇	C₂²×D₄	1	4	yes	—
239	C₂×C₄×C₆₇	—	2	4	no	—
240	C₂×C₄×C₆₇	—	2	4	no	—
241	C₂×C₄×C₆₇	—	2	2	no	—
242	C₂×C₄×C₆₇	—	2	2	no	—
243	C₂×C₄×C₆₇	—	2	8	no	—
244	C₂×C₄×C₆₇	—	2	4	no	—
245	C₂×C₄×C₆₇	—	2	4	no	—
246	C₂×C₄×C₆₇	—	2	2	no	—
247	C₂×C₄×C₆₇	—	2	2	no	—
248	C₂×C₄×C₆₇	—	2	4	no	—
249	C₂×C₄×C₆₇	—	2	4	no	—
250	C₂×C₄×C₆₇	—	4	16	no	—
251	C₂×C₄×C₆₇	—	4	16	no	—
252	C₂×C₄×C₆₇	—	4	4	no	—
253	C₂×C₄×C₆₇	—	4	4	no	—
254	C₂×C₄×C₆₇	—	4	2	no	—
255	C₂×C₄×C₆₇	—	4	2	no	—
256	C₂×C₄×C₆₇	—	4	8	no	—
257	C₂×C₄×C₆₇	#137	1	4	yes	complemented
258	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
259	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
260	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
261	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
262	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
263	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
264	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
265	C₈×C₆₇	—	2	4	no	—
266	C₈×C₆₇	—	2	2	no	—
267	C₈×C₆₇	C₂²×D₄	1	4	yes	—
268	C₂²×C₆₇	—	2	4	no	—
269	C₂²×C₆₇	—	2	4	no	—
270	C₂²×C₆₇	—	2	4	no	—
271	C₂²×C₆₇	—	2	4	no	—
272	C₂²×C₆₇	—	4	8	no	—
273	C₂²×C₆₇	—	4	8	no	—
274	C₂²×C₆₇	—	4	2	no	—
275	C₂²×C₆₇	—	4	2	no	—
276	C₂²×C₆₇	D₄²	1	2	yes	—
277	C₂²×C₆₇	C₂×#137	1	2	yes	—
278	C₂²×C₆₇	#578	1	2	yes	—
279	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
280	C₄×C₆₇	—	2	4	no	—
281	C₄×C₆₇	—	2	4	no	—
282	C₄×C₆₇	—	2	2	no	—
283	C₄×C₆₇	—	4	4	no	—
284	C₄×C₆₇	—	4	4	no	—
285	C₄×C₆₇	—	4	2	no	—
286	C₄×C₆₇	—	4	2	no	—
287	C₄×C₆₇	D₄²	1	2	yes	—
288	C₄×C₆₇	C₂×#137	1	4	yes	complemented
289	C₄×C₆₇	C₂³×D₄	1	1	yes	characteristic
290	C₄×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
291	C₂×#3276	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
292	C₂×C₆₇	—	4	4	no	—
293	C₂×C₆₇	—	4	4	no	—
294	C₂×C₆₇	#3276	1	4	yes	direct product factor
295	C₂×C₆₇	C₂×D₄²	1	1	yes	characteristic
296	C₂×C₆₇	C₂²×#137	1	1	yes	characteristic
297	C₂×C₆₇	C₂×#578	1	1	yes	characteristic
298	C₂²×#137	C₂×C₆₇	1	2	yes	complemented
299	C₂×D₄×Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
300	C₂×#547	C₂×C₆₇	1	1	yes	characteristic
301	#3276	C₂×C₆₇	1	16	yes	direct product factor
302	C₂×#494	C₂×C₆₇	1	1	yes	characteristic, complemented
303	C₂×#470	C₂×C₆₇	1	2	yes	complemented
304	C₂×#462	C₂×C₆₇	1	1	yes	characteristic, complemented
305	C₂×#461	C₂×C₆₇	1	1	yes	characteristic, complemented
306	C₂×#452	C₂×C₆₇	1	2	yes	complemented
307	C₂×#451	C₂×C₆₇	1	2	yes	complemented
308	C₂×C₄×SD₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
309	C₂×#436	C₂×C₆₇	1	1	yes	characteristic, complemented
310	C₆₇	C₂×#3276	1	1	yes	characteristic, direct product factor, Sylow
311	C₂²×#41	C₂²×C₆₇	1	2	yes	—
312	C₂³×Q₈	C₂²×C₆₇	1	2	yes	—
313	C₂×#137	—	2	16	no	—
314	C₂×#137	C₂²×C₆₇	1	16	yes	complemented
315	C₂²×Q₁₆	C₂²×C₆₇	1	2	yes	—
316	C₂²×Q₁₆	C₂²×C₆₇	1	2	yes	—
317	C₂²×Q₁₆	C₂²×C₆₇	1	2	yes	—
318	C₂²×SD₁₆	C₂²×C₆₇	1	2	yes	—
319	C₂²×SD₁₆	C₂²×C₆₇	1	1	yes	characteristic
320	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
321	D₄×Q₈	—	2	4	no	—
322	D₄×Q₈	C₂²×C₆₇	1	8	yes	complemented
323	#547	—	2	4	no	—
324	#547	C₂²×C₆₇	1	8	yes	—
325	C₂×C₄⋊Q₈	—	2	1	no	—
326	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
327	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
328	C₂×#123	—	2	2	no	—
329	C₂×C₂²⋊Q₈	—	2	2	no	—
330	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
331	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
332	C₂×#121	—	2	1	no	—
333	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
334	C₂×C₄×D₄	—	2	1	no	—
335	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
336	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
337	C₂²×C₄⋊C₄	—	2	1	no	—
338	#494	C₂²×C₆₇	1	8	yes	complemented
339	#470	C₂²×C₆₇	1	16	yes	complemented
340	#462	C₂²×C₆₇	1	8	yes	complemented
341	#461	C₂²×C₆₇	1	8	yes	complemented
342	#452	C₂²×C₆₇	1	16	yes	complemented
343	#451	C₂²×C₆₇	1	16	yes	complemented
344	C₄×SD₁₆	C₂²×C₆₇	1	8	yes	complemented
345	#436	C₂²×C₆₇	1	8	yes	complemented
346	C₂×#106	C₂²×C₆₇	1	1	yes	characteristic
347	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
348	C₂×#103	C₂²×C₆₇	1	1	yes	characteristic
349	C₂×#103	C₂²×C₆₇	1	2	yes	—
350	C₂×#103	C₂²×C₆₇	1	2	yes	—
351	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
352	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
353	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
354	C₂×#41	—	2	4	no	—
355	C₂×#41	—	2	8	no	—
356	C₂×#41	—	4	4	no	—
357	C₂×#41	C₂³×C₆₇	1	8	yes	—
358	C₂²×Q₈	—	2	2	no	—
359	C₂²×Q₈	—	2	4	no	—
360	C₂²×Q₈	—	2	8	no	—
361	C₂²×Q₈	—	4	4	no	—
362	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
363	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
364	C₂²×Q₈	C₂³×C₆₇	1	8	yes	—
365	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
366	C₂²×D₄	—	2	2	no	—
367	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
368	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
369	C₂³×C₄	—	2	2	no	—
370	C₂³×C₄	—	2	1	no	—
371	C₂³×C₄	—	2	2	no	—
372	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
373	#137	—	2	64	no	—
374	C₂×Q₁₆	—	2	8	no	—
375	C₂×Q₁₆	—	2	8	no	—
376	C₂×Q₁₆	—	4	4	no	—
377	C₂×Q₁₆	C₂³×C₆₇	1	8	yes	—
378	C₂×Q₁₆	C₂³×C₆₇	1	8	yes	—
379	C₂×Q₁₆	C₂³×C₆₇	1	8	yes	—
380	C₂×SD₁₆	—	2	8	no	—
381	C₂×SD₁₆	D₄×C₆₇	1	8	yes	complemented
382	C₂×SD₁₆	C₂³×C₆₇	1	8	yes	—
383	C₂×SD₁₆	C₂³×C₆₇	1	4	yes	—
384	C₂×M₁₆	—	2	8	no	—
385	C₂×M₁₆	C₂³×C₆₇	1	8	yes	—
386	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
387	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
388	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
389	C₄⋊Q₈	—	2	8	no	—
390	C₄⋊Q₈	—	2	2	no	—
391	C₄⋊Q₈	—	2	2	no	—
392	C₄⋊Q₈	C₂³×C₆₇	1	4	yes	—
393	C₄⋊Q₈	C₂³×C₆₇	1	4	yes	—
394	#123	—	2	16	no	—
395	C₂²⋊Q₈	—	2	4	no	—
396	C₂²⋊Q₈	—	2	4	no	—
397	C₂²⋊Q₈	—	2	16	no	—
398	C₂²⋊Q₈	C₂³×C₆₇	1	8	yes	—
399	C₂²⋊Q₈	C₂³×C₆₇	1	8	yes	—
400	#121	—	2	8	no	—
401	C₄×Q₈	—	2	2	no	—
402	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
403	C₄×D₄	—	2	8	no	—
404	C₄×D₄	—	2	2	no	—
405	C₄×D₄	D₄×C₆₇	1	4	yes	—
406	C₄×D₄	C₂³×C₆₇	1	4	yes	—
407	C₄×D₄	C₂³×C₆₇	1	4	yes	—
408	C₂×C₄⋊C₄	—	2	1	no	—
409	C₂×C₄⋊C₄	—	2	2	no	—
410	C₂×C₄⋊C₄	—	2	1	no	—
411	C₂×C₄⋊C₄	—	2	1	no	—
412	C₂×C₄⋊C₄	—	2	1	no	—
413	C₂×C₄⋊C₄	—	2	2	no	—
414	C₂×C₄⋊C₄	—	2	1	no	—
415	C₂×C₄⋊C₄	—	2	8	no	—
416	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
417	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
418	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
419	C₂×C₂²⋊C₄	—	2	2	no	—
420	C₂×C₂²⋊C₄	—	2	2	no	—
421	C₂×C₂²⋊C₄	—	2	1	no	—
422	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
423	C₂×C₄²	—	2	1	no	—
424	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
425	#106	C₂³×C₆₇	1	4	yes	—
426	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
427	#103	C₂³×C₆₇	1	4	yes	—
428	#103	C₂³×C₆₇	1	8	yes	—
429	#103	C₂³×C₆₇	1	8	yes	—
430	#102	C₂³×C₆₇	1	4	yes	—
431	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	—
432	C₄×C₈	C₂³×C₆₇	1	4	yes	—
433	C₂⁴	—	2	1	no	—
434	C₂⁴	C₂×D₄×C₆₇	1	2	yes	—
435	#41	—	2	16	no	—
436	#41	—	4	8	no	—
437	#41	—	4	16	no	—
438	C₂×Q₈	—	2	1	no	—
439	C₂×Q₈	—	2	8	no	—
440	C₂×Q₈	—	2	2	no	—
441	C₂×Q₈	—	2	2	no	—
442	C₂×Q₈	—	2	2	no	—
443	C₂×Q₈	—	2	16	no	—
444	C₂×Q₈	—	2	4	no	—
445	C₂×Q₈	—	2	4	no	—
446	C₂×Q₈	—	2	4	no	—
447	C₂×Q₈	—	2	4	no	—
448	C₂×Q₈	—	4	8	no	—
449	C₂×Q₈	—	4	8	no	—
450	C₂×Q₈	—	4	16	no	—
451	C₂×Q₈	C₂×D₄×C₆₇	1	4	yes	—
452	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
453	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
454	C₂×Q₈	C₂⁴×C₆₇	1	2	yes	—
455	C₂×D₄	—	2	8	no	—
456	C₂×D₄	—	2	4	no	—
457	C₂×D₄	—	2	1	no	—
458	C₂×D₄	—	2	2	no	—
459	C₂×D₄	—	4	8	no	—
460	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
461	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
462	C₂×D₄	C₂⁴×C₆₇	1	2	yes	—
463	C₂²×C₄	—	2	1	no	—
464	C₂²×C₄	—	2	4	no	—
465	C₂²×C₄	—	2	4	no	—
466	C₂²×C₄	—	2	1	no	—
467	C₂²×C₄	—	2	1	no	—
468	C₂²×C₄	—	2	2	no	—
469	C₂²×C₄	—	2	2	no	—
470	C₂²×C₄	—	2	8	no	—
471	C₂²×C₄	—	2	4	no	—
472	C₂²×C₄	—	2	8	no	—
473	C₂²×C₄	—	4	8	no	—
474	C₂²×C₄	—	4	8	no	—
475	C₂²×C₄	—	4	4	no	—
476	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
477	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
478	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
479	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
480	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
481	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
482	Q₁₆	—	2	16	no	—
483	Q₁₆	—	2	16	no	—
484	Q₁₆	—	4	8	no	—
485	SD₁₆	—	2	16	no	—
486	SD₁₆	C₂×D₄×C₆₇	1	16	yes	complemented
487	M₁₆	—	2	16	no	—
488	C₂×C₈	—	2	4	no	—
489	C₂×C₈	—	2	2	no	—
490	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
491	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
492	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
493	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
494	C₄⋊C₄	—	2	4	no	—
495	C₄⋊C₄	—	2	8	no	—
496	C₄⋊C₄	—	2	4	no	—
497	C₄⋊C₄	—	2	4	no	—
498	C₄⋊C₄	—	2	4	no	—
499	C₄⋊C₄	—	2	8	no	—
500	C₄⋊C₄	—	2	4	no	—
501	C₄⋊C₄	—	2	1	no	—
502	C₄⋊C₄	—	2	1	no	—
503	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
504	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
505	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
506	C₂²⋊C₄	—	2	8	no	—
507	C₂²⋊C₄	—	2	8	no	—
508	C₂²⋊C₄	—	2	4	no	—
509	C₂²⋊C₄	C₂×D₄×C₆₇	1	8	yes	—
510	C₄²	—	2	4	no	—
511	C₄²	C₂×D₄×C₆₇	1	2	yes	—
512	C₄²	C₂⁴×C₆₇	1	2	yes	—
513	C₂³	—	2	4	no	—
514	C₂³	—	2	4	no	—
515	C₂³	—	2	4	no	—
516	C₂³	—	2	4	no	—
517	C₂³	—	2	2	no	—
518	C₂³	—	4	2	no	—
519	C₂³	—	4	2	no	—
520	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
521	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
522	Q₈	—	2	4	no	—
523	Q₈	—	2	2	no	—
524	Q₈	—	2	4	no	—
525	Q₈	—	2	4	no	—
526	Q₈	—	2	4	no	—
527	Q₈	—	2	4	no	—
528	Q₈	—	4	16	no	—
529	Q₈	—	4	4	no	—
530	Q₈	—	4	4	no	—
531	Q₈	C₂²×D₄×C₆₇	1	4	yes	—
532	D₄	—	2	4	no	—
533	D₄	—	2	8	no	—
534	D₄	—	2	2	no	—
535	D₄	—	4	16	no	—
536	D₄	C₂²×D₄×C₆₇	1	4	yes	—
537	C₂×C₄	—	2	4	no	—
538	C₂×C₄	—	2	4	no	—
539	C₂×C₄	—	2	2	no	—
540	C₂×C₄	—	2	2	no	—
541	C₂×C₄	—	2	8	no	—
542	C₂×C₄	—	2	4	no	—
543	C₂×C₄	—	2	4	no	—
544	C₂×C₄	—	2	2	no	—
545	C₂×C₄	—	2	2	no	—
546	C₂×C₄	—	2	4	no	—
547	C₂×C₄	—	2	4	no	—
548	C₂×C₄	—	4	16	no	—
549	C₂×C₄	—	4	16	no	—
550	C₂×C₄	—	4	4	no	—
551	C₂×C₄	—	4	4	no	—
552	C₂×C₄	—	4	2	no	—
553	C₂×C₄	—	4	2	no	—
554	C₂×C₄	—	4	8	no	—
555	C₂×C₄	#137×C₆₇	1	4	yes	complemented
556	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
557	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
558	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
559	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
560	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
561	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
562	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
563	C₈	—	2	4	no	—
564	C₈	—	2	2	no	—
565	C₈	C₂²×D₄×C₆₇	1	4	yes	—
566	C₂²	—	2	4	no	—
567	C₂²	—	2	4	no	—
568	C₂²	—	2	4	no	—
569	C₂²	—	2	4	no	—
570	C₂²	—	4	8	no	—
571	C₂²	—	4	8	no	—
572	C₂²	—	4	2	no	—
573	C₂²	—	4	2	no	—
574	C₂²	D₄²×C₆₇	1	2	yes	—
575	C₂²	C₂×#137×C₆₇	1	2	yes	—
576	C₂²	#578×C₆₇	1	2	yes	—
577	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
578	C₄	—	2	2	no	—
579	C₄	—	2	4	no	—
580	C₄	—	2	4	no	—
581	C₄	—	4	2	no	—
582	C₄	—	4	2	no	—
583	C₄	—	4	4	no	—
584	C₄	—	4	4	no	—
585	C₄	D₄²×C₆₇	1	2	yes	—
586	C₄	C₂×#137×C₆₇	1	4	yes	complemented
587	C₄	C₂³×D₄×C₆₇	1	1	yes	characteristic
588	C₄	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
589	C₂	—	4	4	no	—
590	C₂	—	4	4	no	—
591	C₂	C₆₇×#3276	1	4	yes	direct product factor
592	C₂	C₂×D₄²×C₆₇	1	1	yes	characteristic
593	C₂	C₂²×#137×C₆₇	1	1	yes	characteristic
594	C₂	C₂×#578×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
prime radical	C₂×#547×C₆₇	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×C₄×D₄×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂²×#137×C₆₇	C₂	semidirect
2. C₂×D₄×Q₈×C₆₇	C₂	semidirect
4. C₆₇×#3276	C₂	direct (N × H)
5. C₂×#494×C₆₇	C₂	semidirect
6. C₂×#470×C₆₇	C₂	semidirect
7. C₂×#462×C₆₇	C₂	semidirect
8. C₂×#461×C₆₇	C₂	semidirect
9. C₂×#452×C₆₇	C₂	semidirect
10. C₂×#451×C₆₇	C₂	semidirect
11. C₂×C₄×SD₁₆×C₆₇	C₂	semidirect
12. C₂×#436×C₆₇	C₂	semidirect
16. C₂×#137×C₆₇	C₂²	semidirect
24. D₄×Q₈×C₆₇	C₂²	semidirect
40. #494×C₆₇	C₂²	semidirect
41. #470×C₆₇	C₂²	semidirect
42. #462×C₆₇	C₂²	semidirect
43. #461×C₆₇	C₂²	semidirect
44. #452×C₆₇	C₂²	semidirect
45. #451×C₆₇	C₂²	semidirect
46. C₄×SD₁₆×C₆₇	C₂²	semidirect
47. #436×C₆₇	C₂²	semidirect
83. C₂×SD₁₆×C₆₇	D₄	semidirect
188. SD₁₆×C₆₇	C₂×D₄	semidirect
257. C₂×C₄×C₆₇	#137	semidirect
288. C₄×C₆₇	C₂×#137	semidirect
291. C₂×#3276	C₆₇	direct (N × H)
294. C₂×C₆₇	#3276	direct (N × H)
298. C₂²×#137	C₂×C₆₇	semidirect
299. C₂×D₄×Q₈	C₂×C₆₇	semidirect
301. #3276	C₂×C₆₇	direct (N × H)
302. C₂×#494	C₂×C₆₇	semidirect
303. C₂×#470	C₂×C₆₇	semidirect
304. C₂×#462	C₂×C₆₇	semidirect
305. C₂×#461	C₂×C₆₇	semidirect
306. C₂×#452	C₂×C₆₇	semidirect
307. C₂×#451	C₂×C₆₇	semidirect
308. C₂×C₄×SD₁₆	C₂×C₆₇	semidirect
309. C₂×#436	C₂×C₆₇	semidirect
310. C₆₇	C₂×#3276	direct (N × H)
314. C₂×#137	C₂²×C₆₇	semidirect
322. D₄×Q₈	C₂²×C₆₇	semidirect
338. #494	C₂²×C₆₇	semidirect
339. #470	C₂²×C₆₇	semidirect
340. #462	C₂²×C₆₇	semidirect
341. #461	C₂²×C₆₇	semidirect
342. #452	C₂²×C₆₇	semidirect
343. #451	C₂²×C₆₇	semidirect
344. C₄×SD₁₆	C₂²×C₆₇	semidirect
345. #436	C₂²×C₆₇	semidirect
381. C₂×SD₁₆	D₄×C₆₇	semidirect
486. SD₁₆	C₂×D₄×C₆₇	semidirect
555. C₂×C₄	#137×C₆₇	semidirect
586. C₄	C₂×#137×C₆₇	semidirect
591. C₂	C₆₇×#3276	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×#137×C₆₇	2
C₂×D₄×Q₈×C₆₇	2
C₂×#547×C₆₇	2
C₆₇×#3276	2
C₂×#494×C₆₇	2
C₂×#470×C₆₇	2
C₂×#462×C₆₇	2
C₂×#461×C₆₇	2
C₂×#452×C₆₇	2
C₂×#451×C₆₇	2
C₂×C₄×SD₁₆×C₆₇	2
C₂×#436×C₆₇	2
C₂×#3276	67

Subgroups — C2×C67×#3276 #30740286

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#3276 #30740286