Subgroups — C₂×C₆₇×#3204 #30739999

Order 17152.

2250 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#537×C₆₇	C₂	1	2	yes	maximal, complemented
2	C₆₇×#3204	C₂	1	16	yes	maximal, direct product factor
3	C₂×#481×C₆₇	C₂	1	2	yes	maximal, complemented
4	C₂×#483×C₆₇	C₂	1	2	yes	maximal, complemented
5	C₂×#470×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#469×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#465×C₆₇	C₂	1	2	yes	maximal, complemented
8	C₂×#452×C₆₇	C₂	1	2	yes	maximal, complemented
9	C₂×#449×C₆₇	C₂	1	2	yes	maximal, complemented
10	C₂×#408×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×#41×C₆₇	C₂²	1	2	yes	—
12	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
13	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
14	C₂²×SD₁₆×C₆₇	C₂²	1	2	yes	—
15	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
17	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
18	#537×C₆₇	—	2	8	no	—
19	#537×C₆₇	C₂²	1	16	yes	complemented
20	C₂×C₂²⋊Q₈×C₆₇	—	2	2	no	—
21	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
22	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
23	C₂×#121×C₆₇	—	2	2	no	—
24	C₂×#121×C₆₇	—	2	2	no	—
25	C₂×#121×C₆₇	—	2	2	no	—
26	C₂×#121×C₆₇	C₂²	1	2	yes	—
27	C₂×#121×C₆₇	C₂²	1	2	yes	—
28	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
29	#483×C₆₇	C₂²	1	16	yes	complemented
30	#481×C₆₇	C₂²	1	16	yes	complemented
31	#470×C₆₇	C₂²	1	8	yes	complemented
32	#469×C₆₇	C₂²	1	8	yes	complemented
33	#465×C₆₇	C₂²	1	16	yes	complemented
34	#452×C₆₇	C₂²	1	16	yes	complemented
35	#449×C₆₇	C₂²	1	16	yes	complemented
36	C₂×#106×C₆₇	C₂²	1	2	yes	—
37	C₂×#106×C₆₇	C₂²	1	1	yes	characteristic
38	C₂×#106×C₆₇	C₂²	1	1	yes	characteristic
39	C₂×#103×C₆₇	C₂²	1	2	yes	—
40	C₂×#103×C₆₇	C₂²	1	2	yes	—
41	C₂×#102×C₆₇	C₂²	1	2	yes	—
42	C₂×#102×C₆₇	C₂²	1	2	yes	—
43	#408×C₆₇	C₂²	1	8	yes	complemented
44	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
45	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
46	C₂×#41×C₆₇	—	2	4	no	—
47	C₂×#41×C₆₇	—	4	2	no	—
48	C₂×#41×C₆₇	—	4	2	no	—
49	C₂×#41×C₆₇	—	4	4	no	—
50	C₂×#41×C₆₇	—	4	4	no	—
51	C₂×#41×C₆₇	D₄	1	4	yes	—
52	C₂×#41×C₆₇	C₂³	1	8	yes	—
53	C₂×#41×C₆₇	C₂³	1	4	yes	—
54	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
55	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
56	C₂²×D₄×C₆₇	—	2	2	no	—
57	C₂²×D₄×C₆₇	—	2	2	no	—
58	C₂²×D₄×C₆₇	—	2	2	no	—
59	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
60	C₂²×D₄×C₆₇	D₄	1	2	yes	—
61	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
62	C₂³×C₄×C₆₇	—	2	2	no	—
63	C₂³×C₄×C₆₇	—	2	2	no	—
64	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
65	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
66	C₂×Q₁₆×C₆₇	—	4	2	no	—
67	C₂×Q₁₆×C₆₇	C₂³	1	4	yes	—
68	C₂×SD₁₆×C₆₇	—	4	4	no	—
69	C₂×SD₁₆×C₆₇	C₂³	1	8	yes	—
70	C₂×D₈×C₆₇	—	4	2	no	—
71	C₂×D₈×C₆₇	C₂³	1	4	yes	—
72	C₂×M₁₆×C₆₇	—	4	2	no	—
73	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
74	C₂²×C₈×C₆₇	—	4	2	no	—
75	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
76	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
77	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
78	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
79	C₂²⋊Q₈×C₆₇	—	2	16	no	—
80	C₂²⋊Q₈×C₆₇	—	2	4	no	—
81	C₂²⋊Q₈×C₆₇	—	2	4	no	—
82	C₂²⋊Q₈×C₆₇	C₂³	1	8	yes	—
83	C₂²⋊Q₈×C₆₇	C₂³	1	8	yes	—
84	#121×C₆₇	—	2	16	no	—
85	#121×C₆₇	—	2	16	no	—
86	#121×C₆₇	—	2	16	no	—
87	#121×C₆₇	—	2	4	no	—
88	#121×C₆₇	—	2	4	no	—
89	#121×C₆₇	C₂³	1	8	yes	—
90	#121×C₆₇	C₂³	1	8	yes	—
91	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
92	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
93	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
94	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
95	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
96	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
97	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
98	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
99	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
100	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
101	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
102	#106×C₆₇	C₂³	1	8	yes	—
103	#106×C₆₇	C₂³	1	4	yes	—
104	#106×C₆₇	C₂³	1	4	yes	—
105	#103×C₆₇	C₂³	1	8	yes	—
106	#103×C₆₇	C₂³	1	8	yes	—
107	#102×C₆₇	C₂³	1	8	yes	—
108	#102×C₆₇	C₂³	1	8	yes	—
109	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	—
110	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	—
111	C₂⁴×C₆₇	—	2	2	no	—
112	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
113	C₂⁴×C₆₇	C₂×D₄	1	2	yes	—
114	#41×C₆₇	—	4	8	no	—
115	#41×C₆₇	—	4	4	no	—
116	#41×C₆₇	—	4	4	no	—
117	#41×C₆₇	—	4	8	no	—
118	#41×C₆₇	—	4	8	no	—
119	#41×C₆₇	—	4	16	no	—
120	C₂×Q₈×C₆₇	—	2	2	no	—
121	C₂×Q₈×C₆₇	—	2	4	no	—
122	C₂×Q₈×C₆₇	—	2	4	no	—
123	C₂×Q₈×C₆₇	—	4	2	no	—
124	C₂×Q₈×C₆₇	C₂×D₄	1	4	yes	—
125	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
126	C₂×D₄×C₆₇	—	2	8	no	—
127	C₂×D₄×C₆₇	—	2	8	no	—
128	C₂×D₄×C₆₇	—	2	8	no	—
129	C₂×D₄×C₆₇	—	2	2	no	—
130	C₂×D₄×C₆₇	—	2	4	no	—
131	C₂×D₄×C₆₇	—	2	4	no	—
132	C₂×D₄×C₆₇	—	4	4	no	—
133	C₂×D₄×C₆₇	—	4	2	no	—
134	C₂×D₄×C₆₇	—	4	8	no	—
135	C₂×D₄×C₆₇	—	8	4	no	—
136	C₂×D₄×C₆₇	—	8	4	no	—
137	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
138	C₂×D₄×C₆₇	C₂×D₄	1	8	yes	—
139	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
140	C₂²×C₄×C₆₇	—	2	2	no	—
141	C₂²×C₄×C₆₇	—	2	2	no	—
142	C₂²×C₄×C₆₇	—	2	4	no	—
143	C₂²×C₄×C₆₇	—	2	4	no	—
144	C₂²×C₄×C₆₇	—	2	2	no	—
145	C₂²×C₄×C₆₇	—	2	2	no	—
146	C₂²×C₄×C₆₇	—	2	2	no	—
147	C₂²×C₄×C₆₇	—	2	8	no	—
148	C₂²×C₄×C₆₇	—	2	8	no	—
149	C₂²×C₄×C₆₇	—	4	8	no	—
150	C₂²×C₄×C₆₇	—	8	4	no	—
151	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
152	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
153	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
154	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
155	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
156	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
157	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
158	Q₁₆×C₆₇	—	4	4	no	—
159	SD₁₆×C₆₇	—	4	8	no	—
160	D₈×C₆₇	—	4	4	no	—
161	M₁₆×C₆₇	—	4	4	no	—
162	C₂×C₈×C₆₇	—	2	4	no	—
163	C₂×C₈×C₆₇	—	2	2	no	—
164	C₂×C₈×C₆₇	—	2	2	no	—
165	C₂×C₈×C₆₇	—	4	4	no	—
166	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
167	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
168	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
169	C₄⋊C₄×C₆₇	—	2	8	no	—
170	C₄⋊C₄×C₆₇	—	2	2	no	—
171	C₄⋊C₄×C₆₇	—	2	2	no	—
172	C₄⋊C₄×C₆₇	—	2	8	no	—
173	C₄⋊C₄×C₆₇	—	2	8	no	—
174	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
175	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
176	C₂²⋊C₄×C₆₇	—	2	8	no	—
177	C₂²⋊C₄×C₆₇	—	2	8	no	—
178	C₂²⋊C₄×C₆₇	—	2	8	no	—
179	C₂²⋊C₄×C₆₇	—	2	8	no	—
180	C₂³×C₆₇	—	2	4	no	—
181	C₂³×C₆₇	—	2	2	no	—
182	C₂³×C₆₇	—	2	4	no	—
183	C₂³×C₆₇	—	2	4	no	—
184	C₂³×C₆₇	—	2	4	no	—
185	C₂³×C₆₇	—	2	4	no	—
186	C₂³×C₆₇	—	4	4	no	—
187	C₂³×C₆₇	—	4	4	no	—
188	C₂³×C₆₇	—	4	4	no	—
189	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
190	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
191	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
192	Q₈×C₆₇	—	2	4	no	—
193	Q₈×C₆₇	—	2	4	no	—
194	Q₈×C₆₇	—	4	4	no	—
195	Q₈×C₆₇	—	4	4	no	—
196	D₄×C₆₇	—	2	4	no	—
197	D₄×C₆₇	—	2	4	no	—
198	D₄×C₆₇	—	4	8	no	—
199	D₄×C₆₇	—	4	4	no	—
200	D₄×C₆₇	—	4	16	no	—
201	D₄×C₆₇	—	4	4	no	—
202	D₄×C₆₇	—	8	8	no	—
203	D₄×C₆₇	—	8	8	no	—
204	C₂×C₄×C₆₇	—	2	4	no	—
205	C₂×C₄×C₆₇	—	2	2	no	—
206	C₂×C₄×C₆₇	—	2	4	no	—
207	C₂×C₄×C₆₇	—	2	8	no	—
208	C₂×C₄×C₆₇	—	2	2	no	—
209	C₂×C₄×C₆₇	—	2	2	no	—
210	C₂×C₄×C₆₇	—	2	4	no	—
211	C₂×C₄×C₆₇	—	2	4	no	—
212	C₂×C₄×C₆₇	—	2	4	no	—
213	C₂×C₄×C₆₇	—	2	4	no	—
214	C₂×C₄×C₆₇	—	2	4	no	—
215	C₂×C₄×C₆₇	—	4	16	no	—
216	C₂×C₄×C₆₇	—	4	4	no	—
217	C₂×C₄×C₆₇	—	4	4	no	—
218	C₂×C₄×C₆₇	—	4	4	no	—
219	C₂×C₄×C₆₇	—	8	8	no	—
220	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
221	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
222	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
223	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
224	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
225	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
226	C₈×C₆₇	—	2	4	no	—
227	C₈×C₆₇	—	2	2	no	—
228	C₈×C₆₇	—	2	2	no	—
229	C₂²×C₆₇	—	2	4	no	—
230	C₂²×C₆₇	—	2	2	no	—
231	C₂²×C₆₇	—	2	4	no	—
232	C₂²×C₆₇	—	2	4	no	—
233	C₂²×C₆₇	—	2	4	no	—
234	C₂²×C₆₇	—	4	8	no	—
235	C₂²×C₆₇	—	4	4	no	—
236	C₂²×C₆₇	—	4	4	no	—
237	C₂²×C₆₇	—	4	4	no	—
238	C₂²×C₆₇	—	4	4	no	—
239	C₂²×C₆₇	—	8	8	no	—
240	C₂²×C₆₇	#534	1	2	yes	—
241	C₂²×C₆₇	#577	1	4	yes	—
242	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
243	C₄×C₆₇	—	2	4	no	—
244	C₄×C₆₇	—	2	2	no	—
245	C₄×C₆₇	—	4	4	no	—
246	C₄×C₆₇	—	4	4	no	—
247	C₄×C₆₇	—	4	4	no	—
248	C₄×C₆₇	#534	1	2	yes	—
249	C₄×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
250	C₄×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
251	C₂×#3204	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
252	C₂×C₆₇	—	4	4	no	—
253	C₂×C₆₇	—	4	2	no	—
254	C₂×C₆₇	—	8	4	no	—
255	C₂×C₆₇	#3204	1	4	yes	direct product factor
256	C₂×C₆₇	C₂×#534	1	1	yes	characteristic
257	C₂×C₆₇	C₂×#577	1	2	yes	—
258	C₂×#537	C₂×C₆₇	1	2	yes	complemented
259	#3204	C₂×C₆₇	1	16	yes	direct product factor
260	C₂×#481	C₂×C₆₇	1	2	yes	complemented
261	C₂×#483	C₂×C₆₇	1	2	yes	complemented
262	C₂×#470	C₂×C₆₇	1	1	yes	characteristic, complemented
263	C₂×#469	C₂×C₆₇	1	1	yes	characteristic, complemented
264	C₂×#465	C₂×C₆₇	1	2	yes	complemented
265	C₂×#452	C₂×C₆₇	1	2	yes	complemented
266	C₂×#449	C₂×C₆₇	1	2	yes	complemented
267	C₂×#408	C₂×C₆₇	1	1	yes	characteristic, complemented
268	C₆₇	C₂×#3204	1	1	yes	characteristic, direct product factor, Sylow
269	C₂²×#41	C₂²×C₆₇	1	2	yes	—
270	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
271	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
272	C₂²×SD₁₆	C₂²×C₆₇	1	2	yes	—
273	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
274	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
275	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
276	#537	—	2	8	no	—
277	#537	C₂²×C₆₇	1	16	yes	complemented
278	C₂×C₂²⋊Q₈	—	2	2	no	—
279	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
280	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
281	C₂×#121	—	2	2	no	—
282	C₂×#121	—	2	2	no	—
283	C₂×#121	—	2	2	no	—
284	C₂×#121	C₂²×C₆₇	1	2	yes	—
285	C₂×#121	C₂²×C₆₇	1	2	yes	—
286	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
287	#483	C₂²×C₆₇	1	16	yes	complemented
288	#481	C₂²×C₆₇	1	16	yes	complemented
289	#470	C₂²×C₆₇	1	8	yes	complemented
290	#469	C₂²×C₆₇	1	8	yes	complemented
291	#465	C₂²×C₆₇	1	16	yes	complemented
292	#452	C₂²×C₆₇	1	16	yes	complemented
293	#449	C₂²×C₆₇	1	16	yes	complemented
294	C₂×#106	C₂²×C₆₇	1	1	yes	characteristic
295	C₂×#106	C₂²×C₆₇	1	2	yes	—
296	C₂×#106	C₂²×C₆₇	1	1	yes	characteristic
297	C₂×#103	C₂²×C₆₇	1	2	yes	—
298	C₂×#103	C₂²×C₆₇	1	2	yes	—
299	C₂×#102	C₂²×C₆₇	1	2	yes	—
300	C₂×#102	C₂²×C₆₇	1	2	yes	—
301	#408	C₂²×C₆₇	1	8	yes	complemented
302	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
303	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
304	C₂×#41	—	2	4	no	—
305	C₂×#41	—	4	2	no	—
306	C₂×#41	—	4	2	no	—
307	C₂×#41	—	4	4	no	—
308	C₂×#41	—	4	4	no	—
309	C₂×#41	D₄×C₆₇	1	4	yes	—
310	C₂×#41	C₂³×C₆₇	1	8	yes	—
311	C₂×#41	C₂³×C₆₇	1	4	yes	—
312	C₂²×Q₈	D₄×C₆₇	1	1	yes	characteristic
313	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
314	C₂²×D₄	—	2	2	no	—
315	C₂²×D₄	—	2	2	no	—
316	C₂²×D₄	—	2	2	no	—
317	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
318	C₂²×D₄	D₄×C₆₇	1	2	yes	—
319	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
320	C₂³×C₄	—	2	2	no	—
321	C₂³×C₄	—	2	2	no	—
322	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
323	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
324	C₂×Q₁₆	—	4	2	no	—
325	C₂×Q₁₆	C₂³×C₆₇	1	4	yes	—
326	C₂×SD₁₆	—	4	4	no	—
327	C₂×SD₁₆	C₂³×C₆₇	1	8	yes	—
328	C₂×D₈	—	4	2	no	—
329	C₂×D₈	C₂³×C₆₇	1	4	yes	—
330	C₂×M₁₆	—	4	2	no	—
331	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
332	C₂²×C₈	—	4	2	no	—
333	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
334	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
335	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
336	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
337	C₂²⋊Q₈	—	2	16	no	—
338	C₂²⋊Q₈	—	2	4	no	—
339	C₂²⋊Q₈	—	2	4	no	—
340	C₂²⋊Q₈	C₂³×C₆₇	1	8	yes	—
341	C₂²⋊Q₈	C₂³×C₆₇	1	8	yes	—
342	#121	—	2	16	no	—
343	#121	—	2	16	no	—
344	#121	—	2	16	no	—
345	#121	—	2	4	no	—
346	#121	—	2	4	no	—
347	#121	C₂³×C₆₇	1	8	yes	—
348	#121	C₂³×C₆₇	1	8	yes	—
349	C₂×C₄⋊C₄	—	2	2	no	—
350	C₂×C₄⋊C₄	—	2	4	no	—
351	C₂×C₄⋊C₄	—	2	2	no	—
352	C₂×C₄⋊C₄	—	2	2	no	—
353	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
354	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
355	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
356	C₂×C₂²⋊C₄	—	2	2	no	—
357	C₂×C₂²⋊C₄	—	2	2	no	—
358	C₂×C₂²⋊C₄	—	2	2	no	—
359	C₂×C₂²⋊C₄	—	2	2	no	—
360	#106	C₂³×C₆₇	1	8	yes	—
361	#106	C₂³×C₆₇	1	4	yes	—
362	#106	C₂³×C₆₇	1	4	yes	—
363	#103	C₂³×C₆₇	1	8	yes	—
364	#103	C₂³×C₆₇	1	8	yes	—
365	#102	C₂³×C₆₇	1	8	yes	—
366	#102	C₂³×C₆₇	1	8	yes	—
367	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	—
368	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	—
369	C₂⁴	—	2	2	no	—
370	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
371	C₂⁴	C₂×D₄×C₆₇	1	2	yes	—
372	#41	—	4	8	no	—
373	#41	—	4	4	no	—
374	#41	—	4	4	no	—
375	#41	—	4	8	no	—
376	#41	—	4	8	no	—
377	#41	—	4	16	no	—
378	C₂×Q₈	—	2	2	no	—
379	C₂×Q₈	—	2	4	no	—
380	C₂×Q₈	—	2	4	no	—
381	C₂×Q₈	—	4	2	no	—
382	C₂×Q₈	C₂×D₄×C₆₇	1	4	yes	—
383	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
384	C₂×D₄	—	2	8	no	—
385	C₂×D₄	—	2	8	no	—
386	C₂×D₄	—	2	8	no	—
387	C₂×D₄	—	2	2	no	—
388	C₂×D₄	—	2	4	no	—
389	C₂×D₄	—	2	4	no	—
390	C₂×D₄	—	4	4	no	—
391	C₂×D₄	—	4	2	no	—
392	C₂×D₄	—	4	8	no	—
393	C₂×D₄	—	8	4	no	—
394	C₂×D₄	—	8	4	no	—
395	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
396	C₂×D₄	C₂×D₄×C₆₇	1	8	yes	—
397	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
398	C₂²×C₄	—	2	2	no	—
399	C₂²×C₄	—	2	2	no	—
400	C₂²×C₄	—	2	4	no	—
401	C₂²×C₄	—	2	4	no	—
402	C₂²×C₄	—	2	2	no	—
403	C₂²×C₄	—	2	2	no	—
404	C₂²×C₄	—	2	2	no	—
405	C₂²×C₄	—	2	8	no	—
406	C₂²×C₄	—	2	8	no	—
407	C₂²×C₄	—	4	8	no	—
408	C₂²×C₄	—	8	4	no	—
409	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
410	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
411	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
412	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
413	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
414	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
415	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
416	Q₁₆	—	4	4	no	—
417	SD₁₆	—	4	8	no	—
418	D₈	—	4	4	no	—
419	M₁₆	—	4	4	no	—
420	C₂×C₈	—	2	4	no	—
421	C₂×C₈	—	2	2	no	—
422	C₂×C₈	—	2	2	no	—
423	C₂×C₈	—	4	4	no	—
424	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
425	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
426	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
427	C₄⋊C₄	—	2	8	no	—
428	C₄⋊C₄	—	2	2	no	—
429	C₄⋊C₄	—	2	2	no	—
430	C₄⋊C₄	—	2	8	no	—
431	C₄⋊C₄	—	2	8	no	—
432	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
433	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
434	C₂²⋊C₄	—	2	8	no	—
435	C₂²⋊C₄	—	2	8	no	—
436	C₂²⋊C₄	—	2	8	no	—
437	C₂²⋊C₄	—	2	8	no	—
438	C₂³	—	2	4	no	—
439	C₂³	—	2	2	no	—
440	C₂³	—	2	4	no	—
441	C₂³	—	2	4	no	—
442	C₂³	—	2	4	no	—
443	C₂³	—	2	4	no	—
444	C₂³	—	4	4	no	—
445	C₂³	—	4	4	no	—
446	C₂³	—	4	4	no	—
447	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
448	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
449	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
450	Q₈	—	2	4	no	—
451	Q₈	—	2	4	no	—
452	Q₈	—	4	4	no	—
453	Q₈	—	4	4	no	—
454	D₄	—	2	4	no	—
455	D₄	—	2	4	no	—
456	D₄	—	4	8	no	—
457	D₄	—	4	4	no	—
458	D₄	—	4	16	no	—
459	D₄	—	4	4	no	—
460	D₄	—	8	8	no	—
461	D₄	—	8	8	no	—
462	C₂×C₄	—	2	4	no	—
463	C₂×C₄	—	2	2	no	—
464	C₂×C₄	—	2	8	no	—
465	C₂×C₄	—	2	4	no	—
466	C₂×C₄	—	2	2	no	—
467	C₂×C₄	—	2	2	no	—
468	C₂×C₄	—	2	4	no	—
469	C₂×C₄	—	2	4	no	—
470	C₂×C₄	—	2	4	no	—
471	C₂×C₄	—	2	4	no	—
472	C₂×C₄	—	2	4	no	—
473	C₂×C₄	—	4	16	no	—
474	C₂×C₄	—	4	4	no	—
475	C₂×C₄	—	4	4	no	—
476	C₂×C₄	—	4	4	no	—
477	C₂×C₄	—	8	8	no	—
478	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
479	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
480	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
481	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
482	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
483	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
484	C₈	—	2	4	no	—
485	C₈	—	2	2	no	—
486	C₈	—	2	2	no	—
487	C₂²	—	2	4	no	—
488	C₂²	—	2	4	no	—
489	C₂²	—	2	4	no	—
490	C₂²	—	2	4	no	—
491	C₂²	—	2	2	no	—
492	C₂²	—	4	8	no	—
493	C₂²	—	4	4	no	—
494	C₂²	—	4	4	no	—
495	C₂²	—	4	4	no	—
496	C₂²	—	4	4	no	—
497	C₂²	—	8	8	no	—
498	C₂²	#534×C₆₇	1	2	yes	—
499	C₂²	#577×C₆₇	1	4	yes	—
500	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
501	C₄	—	2	4	no	—
502	C₄	—	2	2	no	—
503	C₄	—	4	4	no	—
504	C₄	—	4	4	no	—
505	C₄	—	4	4	no	—
506	C₄	#534×C₆₇	1	2	yes	—
507	C₄	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
508	C₄	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
509	C₂	—	4	4	no	—
510	C₂	—	4	2	no	—
511	C₂	—	8	4	no	—
512	C₂	C₆₇×#3204	1	4	yes	direct product factor
513	C₂	C₂×#534×C₆₇	1	1	yes	characteristic
514	C₂	C₂×#577×C₆₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂²×#41×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#537×C₆₇	C₂	semidirect
2. C₆₇×#3204	C₂	direct (N × H)
3. C₂×#481×C₆₇	C₂	semidirect
4. C₂×#483×C₆₇	C₂	semidirect
5. C₂×#470×C₆₇	C₂	semidirect
6. C₂×#469×C₆₇	C₂	semidirect
7. C₂×#465×C₆₇	C₂	semidirect
8. C₂×#452×C₆₇	C₂	semidirect
9. C₂×#449×C₆₇	C₂	semidirect
10. C₂×#408×C₆₇	C₂	semidirect
19. #537×C₆₇	C₂²	semidirect
29. #483×C₆₇	C₂²	semidirect
30. #481×C₆₇	C₂²	semidirect
31. #470×C₆₇	C₂²	semidirect
32. #469×C₆₇	C₂²	semidirect
33. #465×C₆₇	C₂²	semidirect
34. #452×C₆₇	C₂²	semidirect
35. #449×C₆₇	C₂²	semidirect
43. #408×C₆₇	C₂²	semidirect
251. C₂×#3204	C₆₇	direct (N × H)
255. C₂×C₆₇	#3204	direct (N × H)
258. C₂×#537	C₂×C₆₇	semidirect
259. #3204	C₂×C₆₇	direct (N × H)
260. C₂×#481	C₂×C₆₇	semidirect
261. C₂×#483	C₂×C₆₇	semidirect
262. C₂×#470	C₂×C₆₇	semidirect
263. C₂×#469	C₂×C₆₇	semidirect
264. C₂×#465	C₂×C₆₇	semidirect
265. C₂×#452	C₂×C₆₇	semidirect
266. C₂×#449	C₂×C₆₇	semidirect
267. C₂×#408	C₂×C₆₇	semidirect
268. C₆₇	C₂×#3204	direct (N × H)
277. #537	C₂²×C₆₇	semidirect
287. #483	C₂²×C₆₇	semidirect
288. #481	C₂²×C₆₇	semidirect
289. #470	C₂²×C₆₇	semidirect
290. #469	C₂²×C₆₇	semidirect
291. #465	C₂²×C₆₇	semidirect
292. #452	C₂²×C₆₇	semidirect
293. #449	C₂²×C₆₇	semidirect
301. #408	C₂²×C₆₇	semidirect
512. C₂	C₆₇×#3204	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×#537×C₆₇	2
C₆₇×#3204	2
C₂×#481×C₆₇	2
C₂×#483×C₆₇	2
C₂×#470×C₆₇	2
C₂×#469×C₆₇	2
C₂×#465×C₆₇	2
C₂×#452×C₆₇	2
C₂×#449×C₆₇	2
C₂×#408×C₆₇	2
C₂×#3204	67

Subgroups — C2×C67×#3204 #30739999

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#3204 #30739999