Subgroups — C₂×C₆₇×#3298 #30740367

Order 17152.

2374 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#547×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#3298	C₂	1	16	yes	maximal, direct product factor
4	C₂×#487×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#483×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₂×#475×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#465×C₆₇	C₂	1	2	yes	maximal, complemented
8	C₂×#461×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#448×C₆₇	C₂	1	2	yes	maximal, complemented
10	C₂×C₄×SD₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₈×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#103×C₆₇	C₂	1	2	yes	maximal, complemented
13	C₂²×#41×C₆₇	—	2	1	no	—
14	C₂³×Q₈×C₆₇	C₂²	1	2	yes	—
15	C₂²×SD₁₆×C₆₇	C₂²	1	1	yes	characteristic
16	C₂²×SD₁₆×C₆₇	C₂²	1	2	yes	—
17	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
18	D₄×Q₈×C₆₇	—	2	4	no	—
19	D₄×Q₈×C₆₇	C₂²	1	8	yes	complemented
20	#547×C₆₇	—	2	4	no	—
21	#547×C₆₇	C₂²	1	8	yes	—
22	C₂×C₄⋊Q₈×C₆₇	—	2	1	no	—
23	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×#123×C₆₇	—	2	2	no	—
26	C₂×C₂²⋊Q₈×C₆₇	—	2	1	no	—
27	C₂×C₂²⋊Q₈×C₆₇	—	2	2	no	—
28	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
29	C₂×#121×C₆₇	C₂²	1	2	yes	—
30	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×C₄×D₄×C₆₇	—	2	1	no	—
32	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
33	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
34	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
35	#487×C₆₇	C₂²	1	8	yes	complemented
36	#483×C₆₇	C₂²	1	16	yes	complemented
37	#475×C₆₇	C₂²	1	8	yes	complemented
38	#465×C₆₇	C₂²	1	16	yes	complemented
39	#461×C₆₇	C₂²	1	8	yes	complemented
40	#448×C₆₇	C₂²	1	16	yes	complemented
41	C₄×SD₁₆×C₆₇	C₂²	1	8	yes	complemented
42	C₈×D₄×C₆₇	C₂²	1	8	yes	complemented
43	C₂×#106×C₆₇	C₂²	1	2	yes	—
44	C₂×#106×C₆₇	C₂²	1	1	yes	characteristic
45	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
46	C₂×#103×C₆₇	C₂²	1	2	yes	—
47	C₂×#103×C₆₇	C₂²	1	2	yes	—
48	C₂×#103×C₆₇	C₂²	1	2	yes	—
49	C₂×#103×C₆₇	C₂²	1	2	yes	—
50	C₂×#103×C₆₇	C₂²	1	1	yes	characteristic
51	C₂×#103×C₆₇	C₂²	1	16	yes	complemented
52	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
53	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
54	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
55	C₂×#41×C₆₇	—	2	8	no	—
56	C₂×#41×C₆₇	—	4	8	no	—
57	C₂²×Q₈×C₆₇	—	2	2	no	—
58	C₂²×Q₈×C₆₇	—	2	1	no	—
59	C₂²×Q₈×C₆₇	—	2	4	no	—
60	C₂²×Q₈×C₆₇	—	2	8	no	—
61	C₂²×Q₈×C₆₇	—	4	4	no	—
62	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
63	C₂²×Q₈×C₆₇	C₂³	1	8	yes	—
64	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
65	C₂²×D₄×C₆₇	—	2	2	no	—
66	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
67	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
68	C₂³×C₄×C₆₇	—	2	2	no	—
69	C₂³×C₄×C₆₇	—	2	2	no	—
70	C₂³×C₄×C₆₇	—	2	1	no	—
71	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
72	C₂×SD₁₆×C₆₇	—	2	4	no	—
73	C₂×SD₁₆×C₆₇	—	4	4	no	—
74	C₂×SD₁₆×C₆₇	C₂³	1	4	yes	—
75	C₂×SD₁₆×C₆₇	C₂³	1	8	yes	—
76	C₂²×C₈×C₆₇	—	4	4	no	—
77	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
78	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
79	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
80	C₂²×C₈×C₆₇	C₂³	1	8	yes	—
81	C₄⋊Q₈×C₆₇	—	2	8	no	—
82	C₄⋊Q₈×C₆₇	—	2	2	no	—
83	C₄⋊Q₈×C₆₇	—	2	2	no	—
84	C₄⋊Q₈×C₆₇	C₂³	1	4	yes	—
85	C₄⋊Q₈×C₆₇	C₂³	1	4	yes	—
86	#123×C₆₇	—	2	16	no	—
87	C₂²⋊Q₈×C₆₇	—	2	4	no	—
88	C₂²⋊Q₈×C₆₇	—	2	8	no	—
89	C₂²⋊Q₈×C₆₇	—	2	16	no	—
90	C₂²⋊Q₈×C₆₇	C₂³	1	8	yes	—
91	#121×C₆₇	—	2	4	no	—
92	#121×C₆₇	C₂³	1	8	yes	—
93	C₄×Q₈×C₆₇	—	2	2	no	—
94	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
95	C₄×D₄×C₆₇	—	2	8	no	—
96	C₄×D₄×C₆₇	—	2	2	no	—
97	C₄×D₄×C₆₇	D₄	1	4	yes	—
98	C₄×D₄×C₆₇	C₂³	1	4	yes	—
99	C₄×D₄×C₆₇	C₂³	1	4	yes	—
100	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
101	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
102	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
103	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
104	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
105	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
106	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
107	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
108	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
109	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
110	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
111	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
112	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
113	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
114	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
115	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
116	C₂×C₄²×C₆₇	—	2	1	no	—
117	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
118	#106×C₆₇	C₂³	1	8	yes	—
119	#106×C₆₇	C₂³	1	4	yes	—
120	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
121	#103×C₆₇	C₂³	1	8	yes	—
122	#103×C₆₇	C₂³	1	8	yes	—
123	#103×C₆₇	C₂³	1	8	yes	—
124	#103×C₆₇	C₂³	1	8	yes	—
125	#103×C₆₇	C₂³	1	4	yes	—
126	#102×C₆₇	C₂³	1	4	yes	—
127	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	—
128	C₄×C₈×C₆₇	C₂³	1	4	yes	—
129	C₂⁴×C₆₇	—	2	1	no	—
130	C₂⁴×C₆₇	C₂×D₄	1	2	yes	—
131	#41×C₆₇	—	4	32	no	—
132	C₂×Q₈×C₆₇	—	2	1	no	—
133	C₂×Q₈×C₆₇	—	2	8	no	—
134	C₂×Q₈×C₆₇	—	2	2	no	—
135	C₂×Q₈×C₆₇	—	2	4	no	—
136	C₂×Q₈×C₆₇	—	2	2	no	—
137	C₂×Q₈×C₆₇	—	2	16	no	—
138	C₂×Q₈×C₆₇	—	2	4	no	—
139	C₂×Q₈×C₆₇	—	2	4	no	—
140	C₂×Q₈×C₆₇	—	4	8	no	—
141	C₂×Q₈×C₆₇	—	4	8	no	—
142	C₂×Q₈×C₆₇	—	4	16	no	—
143	C₂×Q₈×C₆₇	—	4	4	no	—
144	C₂×Q₈×C₆₇	#41	1	4	yes	complemented
145	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
146	C₂×Q₈×C₆₇	C₂⁴	1	2	yes	—
147	C₂×D₄×C₆₇	—	2	8	no	—
148	C₂×D₄×C₆₇	—	2	1	no	—
149	C₂×D₄×C₆₇	—	2	2	no	—
150	C₂×D₄×C₆₇	—	2	2	no	—
151	C₂×D₄×C₆₇	—	4	8	no	—
152	C₂×D₄×C₆₇	SD₁₆	1	8	yes	complemented
153	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
154	C₂×D₄×C₆₇	C₂⁴	1	2	yes	—
155	C₂²×C₄×C₆₇	—	2	1	no	—
156	C₂²×C₄×C₆₇	—	2	4	no	—
157	C₂²×C₄×C₆₇	—	2	4	no	—
158	C₂²×C₄×C₆₇	—	2	8	no	—
159	C₂²×C₄×C₆₇	—	2	8	no	—
160	C₂²×C₄×C₆₇	—	2	1	no	—
161	C₂²×C₄×C₆₇	—	2	2	no	—
162	C₂²×C₄×C₆₇	—	2	1	no	—
163	C₂²×C₄×C₆₇	—	2	2	no	—
164	C₂²×C₄×C₆₇	—	2	4	no	—
165	C₂²×C₄×C₆₇	—	4	8	no	—
166	C₂²×C₄×C₆₇	—	4	8	no	—
167	C₂²×C₄×C₆₇	—	4	4	no	—
168	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
169	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
170	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
171	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
172	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
173	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
174	SD₁₆×C₆₇	—	2	8	no	—
175	SD₁₆×C₆₇	—	4	8	no	—
176	C₂×C₈×C₆₇	—	2	2	no	—
177	C₂×C₈×C₆₇	—	2	2	no	—
178	C₂×C₈×C₆₇	—	2	4	no	—
179	C₂×C₈×C₆₇	—	4	8	no	—
180	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
181	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
182	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
183	C₄⋊C₄×C₆₇	—	2	1	no	—
184	C₄⋊C₄×C₆₇	—	2	2	no	—
185	C₄⋊C₄×C₆₇	—	2	8	no	—
186	C₄⋊C₄×C₆₇	—	2	4	no	—
187	C₄⋊C₄×C₆₇	—	2	4	no	—
188	C₄⋊C₄×C₆₇	—	2	4	no	—
189	C₄⋊C₄×C₆₇	—	2	8	no	—
190	C₄⋊C₄×C₆₇	—	2	4	no	—
191	C₄⋊C₄×C₆₇	—	2	1	no	—
192	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
193	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
194	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
195	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
196	C₂²⋊C₄×C₆₇	—	2	8	no	—
197	C₂²⋊C₄×C₆₇	—	2	4	no	—
198	C₂²⋊C₄×C₆₇	—	2	8	no	—
199	C₂²⋊C₄×C₆₇	C₂×D₄	1	8	yes	—
200	C₄²×C₆₇	—	2	4	no	—
201	C₄²×C₆₇	C₂×D₄	1	2	yes	—
202	C₄²×C₆₇	C₂⁴	1	2	yes	—
203	C₂³×C₆₇	—	2	4	no	—
204	C₂³×C₆₇	—	2	4	no	—
205	C₂³×C₆₇	—	2	2	no	—
206	C₂³×C₆₇	—	4	2	no	—
207	C₂³×C₆₇	—	4	2	no	—
208	C₂³×C₆₇	—	4	2	no	—
209	C₂³×C₆₇	C₂×SD₁₆	1	8	yes	—
210	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
211	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
212	Q₈×C₆₇	—	2	4	no	—
213	Q₈×C₆₇	—	2	2	no	—
214	Q₈×C₆₇	—	2	4	no	—
215	Q₈×C₆₇	—	2	4	no	—
216	Q₈×C₆₇	—	4	16	no	—
217	Q₈×C₆₇	—	4	4	no	—
218	Q₈×C₆₇	—	4	8	no	—
219	Q₈×C₆₇	C₂×#41	1	4	yes	complemented
220	D₄×C₆₇	—	2	2	no	—
221	D₄×C₆₇	—	2	2	no	—
222	D₄×C₆₇	—	4	16	no	—
223	D₄×C₆₇	—	4	2	no	—
224	D₄×C₆₇	C₂×SD₁₆	1	16	yes	complemented
225	C₂×C₄×C₆₇	—	2	4	no	—
226	C₂×C₄×C₆₇	—	2	4	no	—
227	C₂×C₄×C₆₇	—	2	2	no	—
228	C₂×C₄×C₆₇	—	2	2	no	—
229	C₂×C₄×C₆₇	—	2	8	no	—
230	C₂×C₄×C₆₇	—	2	4	no	—
231	C₂×C₄×C₆₇	—	2	4	no	—
232	C₂×C₄×C₆₇	—	2	2	no	—
233	C₂×C₄×C₆₇	—	2	4	no	—
234	C₂×C₄×C₆₇	—	2	2	no	—
235	C₂×C₄×C₆₇	—	2	4	no	—
236	C₂×C₄×C₆₇	—	4	16	no	—
237	C₂×C₄×C₆₇	—	4	4	no	—
238	C₂×C₄×C₆₇	—	4	2	no	—
239	C₂×C₄×C₆₇	—	4	4	no	—
240	C₂×C₄×C₆₇	—	4	16	no	—
241	C₂×C₄×C₆₇	—	4	2	no	—
242	C₂×C₄×C₆₇	—	4	8	no	—
243	C₂×C₄×C₆₇	C₂×SD₁₆	1	4	yes	—
244	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
245	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
246	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
247	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
248	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
249	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
250	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
251	C₈×C₆₇	—	2	2	no	—
252	C₈×C₆₇	—	2	2	no	—
253	C₈×C₆₇	—	2	4	no	—
254	C₂²×C₆₇	—	2	4	no	—
255	C₂²×C₆₇	—	2	16	no	—
256	C₂²×C₆₇	—	2	4	no	—
257	C₂²×C₆₇	—	4	2	no	—
258	C₂²×C₆₇	—	4	2	no	—
259	C₂²×C₆₇	—	4	2	no	—
260	C₂²×C₆₇	—	8	4	no	—
261	C₂²×C₆₇	#546	1	2	yes	—
262	C₂²×C₆₇	C₂²×SD₁₆	1	8	yes	—
263	C₂²×C₆₇	C₂²×SD₁₆	1	2	yes	—
264	C₂²×C₆₇	#578	1	2	yes	—
265	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
266	C₄×C₆₇	—	2	4	no	—
267	C₄×C₆₇	—	2	4	no	—
268	C₄×C₆₇	—	2	2	no	—
269	C₄×C₆₇	—	4	4	no	—
270	C₄×C₆₇	—	4	2	no	—
271	C₄×C₆₇	—	4	4	no	—
272	C₄×C₆₇	—	4	2	no	—
273	C₄×C₆₇	#546	1	2	yes	—
274	C₄×C₆₇	C₂²×SD₁₆	1	4	yes	—
275	C₄×C₆₇	C₂²×#41	1	1	yes	characteristic
276	C₄×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
277	C₂×#3298	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
278	C₂×C₆₇	—	2	8	no	—
279	C₂×C₆₇	—	8	2	no	—
280	C₂×C₆₇	#3298	1	4	yes	direct product factor
281	C₂×C₆₇	C₂×#546	1	1	yes	characteristic
282	C₂×C₆₇	C₂³×SD₁₆	1	1	yes	characteristic
283	C₂×C₆₇	C₂×#578	1	1	yes	characteristic
284	C₂×D₄×Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
285	C₂×#547	C₂×C₆₇	1	1	yes	characteristic
286	#3298	C₂×C₆₇	1	16	yes	direct product factor
287	C₂×#487	C₂×C₆₇	1	1	yes	characteristic, complemented
288	C₂×#483	C₂×C₆₇	1	2	yes	complemented
289	C₂×#475	C₂×C₆₇	1	1	yes	characteristic, complemented
290	C₂×#465	C₂×C₆₇	1	2	yes	complemented
291	C₂×#461	C₂×C₆₇	1	1	yes	characteristic, complemented
292	C₂×#448	C₂×C₆₇	1	2	yes	complemented
293	C₂×C₄×SD₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
294	C₂×C₈×D₄	C₂×C₆₇	1	1	yes	characteristic, complemented
295	C₂²×#103	C₂×C₆₇	1	2	yes	complemented
296	C₆₇	C₂×#3298	1	1	yes	characteristic, direct product factor, Sylow
297	C₂²×#41	—	2	1	no	—
298	C₂³×Q₈	C₂²×C₆₇	1	2	yes	—
299	C₂²×SD₁₆	C₂²×C₆₇	1	1	yes	characteristic
300	C₂²×SD₁₆	C₂²×C₆₇	1	2	yes	—
301	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
302	D₄×Q₈	—	2	4	no	—
303	D₄×Q₈	C₂²×C₆₇	1	8	yes	complemented
304	#547	—	2	4	no	—
305	#547	C₂²×C₆₇	1	8	yes	—
306	C₂×C₄⋊Q₈	—	2	1	no	—
307	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
308	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
309	C₂×#123	—	2	2	no	—
310	C₂×C₂²⋊Q₈	—	2	1	no	—
311	C₂×C₂²⋊Q₈	—	2	2	no	—
312	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
313	C₂×#121	C₂²×C₆₇	1	2	yes	—
314	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
315	C₂×C₄×D₄	—	2	1	no	—
316	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
317	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
318	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
319	#487	C₂²×C₆₇	1	8	yes	complemented
320	#483	C₂²×C₆₇	1	16	yes	complemented
321	#475	C₂²×C₆₇	1	8	yes	complemented
322	#465	C₂²×C₆₇	1	16	yes	complemented
323	#461	C₂²×C₆₇	1	8	yes	complemented
324	#448	C₂²×C₆₇	1	16	yes	complemented
325	C₄×SD₁₆	C₂²×C₆₇	1	8	yes	complemented
326	C₈×D₄	C₂²×C₆₇	1	8	yes	complemented
327	C₂×#106	C₂²×C₆₇	1	2	yes	—
328	C₂×#106	C₂²×C₆₇	1	1	yes	characteristic
329	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
330	C₂×#103	C₂²×C₆₇	1	2	yes	—
331	C₂×#103	C₂²×C₆₇	1	2	yes	—
332	C₂×#103	C₂²×C₆₇	1	2	yes	—
333	C₂×#103	C₂²×C₆₇	1	2	yes	—
334	C₂×#103	C₂²×C₆₇	1	1	yes	characteristic
335	C₂×#103	C₂²×C₆₇	1	16	yes	complemented
336	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
337	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
338	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
339	C₂×#41	—	2	8	no	—
340	C₂×#41	—	4	8	no	—
341	C₂²×Q₈	—	2	2	no	—
342	C₂²×Q₈	—	2	1	no	—
343	C₂²×Q₈	—	2	4	no	—
344	C₂²×Q₈	—	2	8	no	—
345	C₂²×Q₈	—	4	4	no	—
346	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
347	C₂²×Q₈	C₂³×C₆₇	1	8	yes	—
348	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
349	C₂²×D₄	—	2	2	no	—
350	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
351	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
352	C₂³×C₄	—	2	2	no	—
353	C₂³×C₄	—	2	2	no	—
354	C₂³×C₄	—	2	1	no	—
355	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
356	C₂×SD₁₆	—	2	4	no	—
357	C₂×SD₁₆	—	4	4	no	—
358	C₂×SD₁₆	C₂³×C₆₇	1	4	yes	—
359	C₂×SD₁₆	C₂³×C₆₇	1	8	yes	—
360	C₂²×C₈	—	4	4	no	—
361	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
362	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
363	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
364	C₂²×C₈	C₂³×C₆₇	1	8	yes	—
365	C₄⋊Q₈	—	2	8	no	—
366	C₄⋊Q₈	—	2	2	no	—
367	C₄⋊Q₈	—	2	2	no	—
368	C₄⋊Q₈	C₂³×C₆₇	1	4	yes	—
369	C₄⋊Q₈	C₂³×C₆₇	1	4	yes	—
370	#123	—	2	16	no	—
371	C₂²⋊Q₈	—	2	4	no	—
372	C₂²⋊Q₈	—	2	8	no	—
373	C₂²⋊Q₈	—	2	16	no	—
374	C₂²⋊Q₈	C₂³×C₆₇	1	8	yes	—
375	#121	—	2	4	no	—
376	#121	C₂³×C₆₇	1	8	yes	—
377	C₄×Q₈	—	2	2	no	—
378	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
379	C₄×D₄	—	2	8	no	—
380	C₄×D₄	—	2	2	no	—
381	C₄×D₄	D₄×C₆₇	1	4	yes	—
382	C₄×D₄	C₂³×C₆₇	1	4	yes	—
383	C₄×D₄	C₂³×C₆₇	1	4	yes	—
384	C₂×C₄⋊C₄	—	2	2	no	—
385	C₂×C₄⋊C₄	—	2	4	no	—
386	C₂×C₄⋊C₄	—	2	1	no	—
387	C₂×C₄⋊C₄	—	2	1	no	—
388	C₂×C₄⋊C₄	—	2	1	no	—
389	C₂×C₄⋊C₄	—	2	2	no	—
390	C₂×C₄⋊C₄	—	2	1	no	—
391	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
392	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
393	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
394	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
395	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
396	C₂×C₂²⋊C₄	—	2	2	no	—
397	C₂×C₂²⋊C₄	—	2	1	no	—
398	C₂×C₂²⋊C₄	—	2	2	no	—
399	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
400	C₂×C₄²	—	2	1	no	—
401	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
402	#106	C₂³×C₆₇	1	4	yes	—
403	#106	C₂³×C₆₇	1	8	yes	—
404	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
405	#103	C₂³×C₆₇	1	4	yes	—
406	#103	C₂³×C₆₇	1	8	yes	—
407	#103	C₂³×C₆₇	1	8	yes	—
408	#103	C₂³×C₆₇	1	8	yes	—
409	#103	C₂³×C₆₇	1	8	yes	—
410	#102	C₂³×C₆₇	1	4	yes	—
411	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	—
412	C₄×C₈	C₂³×C₆₇	1	4	yes	—
413	C₂⁴	—	2	1	no	—
414	C₂⁴	C₂×D₄×C₆₇	1	2	yes	—
415	#41	—	4	32	no	—
416	C₂×Q₈	—	2	1	no	—
417	C₂×Q₈	—	2	8	no	—
418	C₂×Q₈	—	2	2	no	—
419	C₂×Q₈	—	2	4	no	—
420	C₂×Q₈	—	2	2	no	—
421	C₂×Q₈	—	2	16	no	—
422	C₂×Q₈	—	2	4	no	—
423	C₂×Q₈	—	2	4	no	—
424	C₂×Q₈	—	4	8	no	—
425	C₂×Q₈	—	4	8	no	—
426	C₂×Q₈	—	4	16	no	—
427	C₂×Q₈	—	4	4	no	—
428	C₂×Q₈	#41×C₆₇	1	4	yes	complemented
429	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
430	C₂×Q₈	C₂⁴×C₆₇	1	2	yes	—
431	C₂×D₄	—	2	8	no	—
432	C₂×D₄	—	2	1	no	—
433	C₂×D₄	—	2	2	no	—
434	C₂×D₄	—	2	2	no	—
435	C₂×D₄	—	4	8	no	—
436	C₂×D₄	SD₁₆×C₆₇	1	8	yes	complemented
437	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
438	C₂×D₄	C₂⁴×C₆₇	1	2	yes	—
439	C₂²×C₄	—	2	1	no	—
440	C₂²×C₄	—	2	4	no	—
441	C₂²×C₄	—	2	4	no	—
442	C₂²×C₄	—	2	1	no	—
443	C₂²×C₄	—	2	2	no	—
444	C₂²×C₄	—	2	1	no	—
445	C₂²×C₄	—	2	2	no	—
446	C₂²×C₄	—	2	8	no	—
447	C₂²×C₄	—	2	8	no	—
448	C₂²×C₄	—	2	4	no	—
449	C₂²×C₄	—	4	8	no	—
450	C₂²×C₄	—	4	8	no	—
451	C₂²×C₄	—	4	4	no	—
452	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
453	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
454	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
455	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
456	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
457	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
458	SD₁₆	—	2	8	no	—
459	SD₁₆	—	4	8	no	—
460	C₂×C₈	—	2	2	no	—
461	C₂×C₈	—	2	2	no	—
462	C₂×C₈	—	2	4	no	—
463	C₂×C₈	—	4	8	no	—
464	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
465	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
466	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
467	C₄⋊C₄	—	2	1	no	—
468	C₄⋊C₄	—	2	2	no	—
469	C₄⋊C₄	—	2	8	no	—
470	C₄⋊C₄	—	2	4	no	—
471	C₄⋊C₄	—	2	4	no	—
472	C₄⋊C₄	—	2	4	no	—
473	C₄⋊C₄	—	2	8	no	—
474	C₄⋊C₄	—	2	4	no	—
475	C₄⋊C₄	—	2	1	no	—
476	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
477	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
478	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
479	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
480	C₂²⋊C₄	—	2	8	no	—
481	C₂²⋊C₄	—	2	4	no	—
482	C₂²⋊C₄	—	2	8	no	—
483	C₂²⋊C₄	C₂×D₄×C₆₇	1	8	yes	—
484	C₄²	—	2	4	no	—
485	C₄²	C₂×D₄×C₆₇	1	2	yes	—
486	C₄²	C₂⁴×C₆₇	1	2	yes	—
487	C₂³	—	2	4	no	—
488	C₂³	—	2	4	no	—
489	C₂³	—	2	2	no	—
490	C₂³	—	4	2	no	—
491	C₂³	—	4	2	no	—
492	C₂³	—	4	2	no	—
493	C₂³	C₂×SD₁₆×C₆₇	1	8	yes	—
494	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
495	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
496	Q₈	—	2	4	no	—
497	Q₈	—	2	2	no	—
498	Q₈	—	2	4	no	—
499	Q₈	—	2	4	no	—
500	Q₈	—	4	16	no	—
501	Q₈	—	4	4	no	—
502	Q₈	—	4	8	no	—
503	Q₈	C₂×#41×C₆₇	1	4	yes	complemented
504	D₄	—	2	2	no	—
505	D₄	—	2	2	no	—
506	D₄	—	4	16	no	—
507	D₄	—	4	2	no	—
508	D₄	C₂×SD₁₆×C₆₇	1	16	yes	complemented
509	C₂×C₄	—	2	4	no	—
510	C₂×C₄	—	2	4	no	—
511	C₂×C₄	—	2	2	no	—
512	C₂×C₄	—	2	2	no	—
513	C₂×C₄	—	2	8	no	—
514	C₂×C₄	—	2	4	no	—
515	C₂×C₄	—	2	4	no	—
516	C₂×C₄	—	2	2	no	—
517	C₂×C₄	—	2	4	no	—
518	C₂×C₄	—	2	2	no	—
519	C₂×C₄	—	2	4	no	—
520	C₂×C₄	—	4	16	no	—
521	C₂×C₄	—	4	4	no	—
522	C₂×C₄	—	4	2	no	—
523	C₂×C₄	—	4	4	no	—
524	C₂×C₄	—	4	16	no	—
525	C₂×C₄	—	4	2	no	—
526	C₂×C₄	—	4	8	no	—
527	C₂×C₄	C₂×SD₁₆×C₆₇	1	4	yes	—
528	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
529	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
530	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
531	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
532	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
533	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
534	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
535	C₈	—	2	2	no	—
536	C₈	—	2	2	no	—
537	C₈	—	2	4	no	—
538	C₂²	—	2	16	no	—
539	C₂²	—	2	4	no	—
540	C₂²	—	2	4	no	—
541	C₂²	—	4	2	no	—
542	C₂²	—	4	2	no	—
543	C₂²	—	4	2	no	—
544	C₂²	—	8	4	no	—
545	C₂²	#546×C₆₇	1	2	yes	—
546	C₂²	C₂²×SD₁₆×C₆₇	1	8	yes	—
547	C₂²	C₂²×SD₁₆×C₆₇	1	2	yes	—
548	C₂²	#578×C₆₇	1	2	yes	—
549	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
550	C₄	—	2	2	no	—
551	C₄	—	2	4	no	—
552	C₄	—	2	4	no	—
553	C₄	—	4	2	no	—
554	C₄	—	4	4	no	—
555	C₄	—	4	2	no	—
556	C₄	—	4	4	no	—
557	C₄	#546×C₆₇	1	2	yes	—
558	C₄	C₂²×SD₁₆×C₆₇	1	4	yes	—
559	C₄	C₂²×#41×C₆₇	1	1	yes	characteristic
560	C₄	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
561	C₂	—	2	8	no	—
562	C₂	—	8	2	no	—
563	C₂	C₆₇×#3298	1	4	yes	direct product factor
564	C₂	C₂×#546×C₆₇	1	1	yes	characteristic
565	C₂	C₂³×SD₁₆×C₆₇	1	1	yes	characteristic
566	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. C₂×D₄×Q₈×C₆₇	C₂	semidirect
3. C₆₇×#3298	C₂	direct (N × H)
4. C₂×#487×C₆₇	C₂	semidirect
5. C₂×#483×C₆₇	C₂	semidirect
6. C₂×#475×C₆₇	C₂	semidirect
7. C₂×#465×C₆₇	C₂	semidirect
8. C₂×#461×C₆₇	C₂	semidirect
9. C₂×#448×C₆₇	C₂	semidirect
10. C₂×C₄×SD₁₆×C₆₇	C₂	semidirect
11. C₂×C₈×D₄×C₆₇	C₂	semidirect
12. C₂²×#103×C₆₇	C₂	semidirect
19. D₄×Q₈×C₆₇	C₂²	semidirect
35. #487×C₆₇	C₂²	semidirect
36. #483×C₆₇	C₂²	semidirect
37. #475×C₆₇	C₂²	semidirect
38. #465×C₆₇	C₂²	semidirect
39. #461×C₆₇	C₂²	semidirect
40. #448×C₆₇	C₂²	semidirect
41. C₄×SD₁₆×C₆₇	C₂²	semidirect
42. C₈×D₄×C₆₇	C₂²	semidirect
51. C₂×#103×C₆₇	C₂²	semidirect
144. C₂×Q₈×C₆₇	#41	semidirect
152. C₂×D₄×C₆₇	SD₁₆	semidirect
219. Q₈×C₆₇	C₂×#41	semidirect
224. D₄×C₆₇	C₂×SD₁₆	semidirect
277. C₂×#3298	C₆₇	direct (N × H)
280. C₂×C₆₇	#3298	direct (N × H)
284. C₂×D₄×Q₈	C₂×C₆₇	semidirect
286. #3298	C₂×C₆₇	direct (N × H)
287. C₂×#487	C₂×C₆₇	semidirect
288. C₂×#483	C₂×C₆₇	semidirect
289. C₂×#475	C₂×C₆₇	semidirect
290. C₂×#465	C₂×C₆₇	semidirect
291. C₂×#461	C₂×C₆₇	semidirect
292. C₂×#448	C₂×C₆₇	semidirect
293. C₂×C₄×SD₁₆	C₂×C₆₇	semidirect
294. C₂×C₈×D₄	C₂×C₆₇	semidirect
295. C₂²×#103	C₂×C₆₇	semidirect
296. C₆₇	C₂×#3298	direct (N × H)
303. D₄×Q₈	C₂²×C₆₇	semidirect
319. #487	C₂²×C₆₇	semidirect
320. #483	C₂²×C₆₇	semidirect
321. #475	C₂²×C₆₇	semidirect
322. #465	C₂²×C₆₇	semidirect
323. #461	C₂²×C₆₇	semidirect
324. #448	C₂²×C₆₇	semidirect
325. C₄×SD₁₆	C₂²×C₆₇	semidirect
326. C₈×D₄	C₂²×C₆₇	semidirect
335. C₂×#103	C₂²×C₆₇	semidirect
428. C₂×Q₈	#41×C₆₇	semidirect
436. C₂×D₄	SD₁₆×C₆₇	semidirect
503. Q₈	C₂×#41×C₆₇	semidirect
508. D₄	C₂×SD₁₆×C₆₇	semidirect
563. C₂	C₆₇×#3298	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×D₄×Q₈×C₆₇	2
C₂×#547×C₆₇	2
C₆₇×#3298	2
C₂×#487×C₆₇	2
C₂×#483×C₆₇	2
C₂×#475×C₆₇	2
C₂×#465×C₆₇	2
C₂×#461×C₆₇	2
C₂×#448×C₆₇	2
C₂×C₄×SD₁₆×C₆₇	2
C₂×C₈×D₄×C₆₇	2
C₂²×#103×C₆₇	2
C₂×#3298	67

Subgroups — C2×C67×#3298 #30740367

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#3298 #30740367