Subgroups — C₆₇×#32942 #30712193

Order 17152.

1368 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#131×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#472×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₂×#466×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#3006	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1907	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1900	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1816	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1810	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
10	C₂×AGL(1,ℤ/8)×C₆₇	C₂²	1	4	yes	—
11	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
12	C₂×#131×C₆₇	—	2	8	no	—
13	C₂×#131×C₆₇	—	2	1	no	—
14	C₂×#131×C₆₇	—	2	1	no	—
15	C₂×#131×C₆₇	C₂²	1	2	yes	—
16	C₂×#131×C₆₇	C₂²	1	2	yes	—
17	C₂²×M₁₆×C₆₇	—	2	1	no	—
18	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
19	C₂³×C₈×C₆₇	—	2	1	no	—
20	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×#121×C₆₇	C₂²	1	2	yes	—
22	#472×C₆₇	—	2	8	no	—
23	#466×C₆₇	—	2	1	no	—
24	#466×C₆₇	—	2	1	no	—
25	#466×C₆₇	—	2	1	no	—
26	#466×C₆₇	—	2	1	no	—
27	#449×C₆₇	—	2	4	no	—
28	#448×C₆₇	—	2	2	no	—
29	#447×C₆₇	—	2	2	no	—
30	C₂×#108×C₆₇	C₂²	1	2	yes	complemented
31	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic, complemented
32	#417×C₆₇	C₂²	1	4	yes	—
33	C₂×#102×C₆₇	C₂²	1	2	yes	complemented
34	C₂×#100×C₆₇	C₂²	1	4	yes	complemented
35	#407×C₆₇	C₂²	1	2	yes	—
36	#343×C₆₇	C₂²	1	4	yes	complemented
37	#341×C₆₇	C₂²	1	2	yes	complemented
38	C₂×#41×C₆₇	—	2	4	no	—
39	C₂×#41×C₆₇	—	2	4	no	—
40	C₂×#41×C₆₇	—	2	1	no	—
41	C₂×#41×C₆₇	—	2	1	no	—
42	C₂×#41×C₆₇	D₄	1	2	yes	—
43	C₂×#41×C₆₇	C₂³	1	2	yes	—
44	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
45	C₂²×D₄×C₆₇	—	2	1	no	—
46	C₂²×D₄×C₆₇	—	2	2	no	—
47	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
48	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
49	C₂³×C₄×C₆₇	—	2	1	no	—
50	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
51	AGL(1,ℤ/8)×C₆₇	—	2	8	no	—
52	AGL(1,ℤ/8)×C₆₇	—	2	8	no	—
53	C₂×SD₁₆×C₆₇	—	2	4	no	—
54	C₂×D₈×C₆₇	—	2	4	no	—
55	C₂×D₈×C₆₇	—	2	1	no	—
56	C₂×D₈×C₆₇	—	2	1	no	—
57	C₂×D₈×C₆₇	—	2	4	no	—
58	#131×C₆₇	—	2	8	no	—
59	#131×C₆₇	—	2	8	no	—
60	#131×C₆₇	—	2	4	no	—
61	#131×C₆₇	—	2	4	no	—
62	#131×C₆₇	—	2	4	no	—
63	#131×C₆₇	D₄	1	8	yes	complemented
64	C₂×M₁₆×C₆₇	—	2	8	no	—
65	C₂×M₁₆×C₆₇	—	2	1	no	—
66	C₂×M₁₆×C₆₇	—	2	4	no	—
67	C₂×M₁₆×C₆₇	—	2	2	no	—
68	C₂×M₁₆×C₆₇	—	2	2	no	—
69	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
70	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
71	C₂²×C₈×C₆₇	—	2	8	no	—
72	C₂²×C₈×C₆₇	—	2	1	no	—
73	C₂²×C₈×C₆₇	—	2	2	no	—
74	C₂²×C₈×C₆₇	—	2	2	no	—
75	C₂²×C₈×C₆₇	—	2	4	no	—
76	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
77	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
78	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
79	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
80	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
81	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
82	#121×C₆₇	—	2	2	no	—
83	#121×C₆₇	—	2	2	no	—
84	#121×C₆₇	—	2	2	no	—
85	#121×C₆₇	—	2	2	no	—
86	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
87	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
88	#108×C₆₇	—	2	4	no	—
89	#107×C₆₇	—	2	1	no	—
90	#107×C₆₇	—	2	1	no	—
91	#103×C₆₇	—	2	4	no	—
92	#102×C₆₇	—	2	4	no	—
93	#102×C₆₇	—	2	2	no	—
94	#102×C₆₇	—	2	2	no	—
95	#100×C₆₇	—	2	8	no	—
96	C₂²⋊C₈×C₆₇	—	2	2	no	—
97	C₂²⋊C₈×C₆₇	—	2	2	no	—
98	C₂⁴×C₆₇	—	2	1	no	—
99	C₂⁴×C₆₇	—	2	2	no	—
100	C₂⁴×C₆₇	#41	1	1	yes	characteristic
101	#41×C₆₇	—	2	4	no	—
102	#41×C₆₇	—	2	8	no	—
103	#41×C₆₇	—	2	4	no	—
104	#41×C₆₇	—	2	4	no	—
105	#41×C₆₇	—	2	4	no	—
106	#41×C₆₇	—	2	2	no	—
107	#41×C₆₇	—	2	2	no	—
108	#41×C₆₇	C₂×D₄	1	4	yes	—
109	#41×C₆₇	#41	1	4	yes	—
110	C₂×Q₈×C₆₇	—	2	4	no	—
111	C₂×Q₈×C₆₇	—	2	1	no	—
112	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
113	C₂×D₄×C₆₇	—	2	8	no	—
114	C₂×D₄×C₆₇	—	2	2	no	—
115	C₂×D₄×C₆₇	—	2	2	no	—
116	C₂×D₄×C₆₇	—	2	4	no	—
117	C₂×D₄×C₆₇	—	2	1	no	—
118	C₂×D₄×C₆₇	—	2	4	no	—
119	C₂×D₄×C₆₇	—	2	2	no	—
120	C₂×D₄×C₆₇	—	2	2	no	—
121	C₂×D₄×C₆₇	—	2	4	no	—
122	C₂×D₄×C₆₇	—	4	2	no	—
123	C₂×D₄×C₆₇	—	4	2	no	—
124	C₂×D₄×C₆₇	—	8	4	no	—
125	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
126	C₂²×C₄×C₆₇	—	2	4	no	—
127	C₂²×C₄×C₆₇	—	2	4	no	—
128	C₂²×C₄×C₆₇	—	2	1	no	—
129	C₂²×C₄×C₆₇	—	2	2	no	—
130	C₂²×C₄×C₆₇	—	2	2	no	—
131	C₂²×C₄×C₆₇	—	2	2	no	—
132	C₂²×C₄×C₆₇	—	2	2	no	—
133	C₂²×C₄×C₆₇	—	2	2	no	—
134	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
135	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
136	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
137	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
138	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
139	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
140	SD₁₆×C₆₇	—	4	8	no	—
141	D₈×C₆₇	—	4	8	no	—
142	D₈×C₆₇	—	4	2	no	—
143	D₈×C₆₇	—	4	2	no	—
144	M₁₆×C₆₇	—	2	4	no	—
145	M₁₆×C₆₇	—	2	8	no	—
146	M₁₆×C₆₇	—	2	2	no	—
147	M₁₆×C₆₇	—	2	8	no	—
148	M₁₆×C₆₇	—	2	4	no	—
149	M₁₆×C₆₇	—	2	4	no	—
150	M₁₆×C₆₇	C₂×D₄	1	4	yes	—
151	C₂×C₈×C₆₇	—	2	4	no	—
152	C₂×C₈×C₆₇	—	2	8	no	—
153	C₂×C₈×C₆₇	—	2	2	no	—
154	C₂×C₈×C₆₇	—	2	2	no	—
155	C₂×C₈×C₆₇	—	2	2	no	—
156	C₂×C₈×C₆₇	—	2	8	no	—
157	C₂×C₈×C₆₇	—	2	2	no	—
158	C₂×C₈×C₆₇	—	2	2	no	—
159	C₂×C₈×C₆₇	—	2	2	no	—
160	C₂×C₈×C₆₇	—	2	4	no	—
161	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
162	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
163	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
164	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
165	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
166	C₄⋊C₄×C₆₇	—	2	2	no	—
167	C₄⋊C₄×C₆₇	—	2	2	no	—
168	C₂²⋊C₄×C₆₇	—	4	2	no	—
169	C₂²⋊C₄×C₆₇	—	4	2	no	—
170	C₂³×C₆₇	—	2	4	no	—
171	C₂³×C₆₇	—	2	2	no	—
172	C₂³×C₆₇	—	2	2	no	—
173	C₂³×C₆₇	—	2	2	no	—
174	C₂³×C₆₇	—	4	1	no	—
175	C₂³×C₆₇	—	4	2	no	—
176	C₂³×C₆₇	—	4	2	no	—
177	C₂³×C₆₇	—	4	2	no	—
178	C₂³×C₆₇	—	4	4	no	—
179	C₂³×C₆₇	—	8	2	no	—
180	C₂³×C₆₇	—	8	2	no	—
181	C₂³×C₆₇	#121	1	1	yes	characteristic
182	C₂³×C₆₇	#121	1	1	yes	characteristic
183	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
184	Q₈×C₆₇	—	2	4	no	—
185	Q₈×C₆₇	—	2	2	no	—
186	Q₈×C₆₇	#121	1	4	yes	—
187	D₄×C₆₇	—	2	8	no	—
188	D₄×C₆₇	—	2	4	no	—
189	D₄×C₆₇	—	2	4	no	—
190	D₄×C₆₇	—	2	4	no	—
191	D₄×C₆₇	—	2	2	no	—
192	D₄×C₆₇	—	4	2	no	—
193	D₄×C₆₇	—	4	4	no	—
194	D₄×C₆₇	—	4	2	no	—
195	D₄×C₆₇	—	8	4	no	—
196	D₄×C₆₇	—	8	4	no	—
197	D₄×C₆₇	#121	1	4	yes	—
198	C₂×C₄×C₆₇	—	2	4	no	—
199	C₂×C₄×C₆₇	—	2	8	no	—
200	C₂×C₄×C₆₇	—	2	2	no	—
201	C₂×C₄×C₆₇	—	2	2	no	—
202	C₂×C₄×C₆₇	—	2	1	no	—
203	C₂×C₄×C₆₇	—	2	4	no	—
204	C₂×C₄×C₆₇	—	2	1	no	—
205	C₂×C₄×C₆₇	—	2	2	no	—
206	C₂×C₄×C₆₇	—	2	2	no	—
207	C₂×C₄×C₆₇	—	2	1	no	—
208	C₂×C₄×C₆₇	—	2	4	no	—
209	C₂×C₄×C₆₇	—	2	1	no	—
210	C₂×C₄×C₆₇	—	4	2	no	—
211	C₂×C₄×C₆₇	—	8	2	no	—
212	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
213	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
214	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
215	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
216	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
217	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
218	C₈×C₆₇	—	2	1	no	—
219	C₈×C₆₇	—	2	4	no	—
220	C₈×C₆₇	—	2	1	no	—
221	C₈×C₆₇	—	4	4	no	—
222	C₈×C₆₇	#120	1	2	yes	—
223	C₈×C₆₇	#120	1	2	yes	—
224	C₂²×C₆₇	—	2	1	no	—
225	C₂²×C₆₇	—	2	1	no	—
226	C₂²×C₆₇	—	2	1	no	—
227	C₂²×C₆₇	—	2	1	no	—
228	C₂²×C₆₇	—	2	1	no	—
229	C₂²×C₆₇	—	2	4	no	—
230	C₂²×C₆₇	—	2	1	no	—
231	C₂²×C₆₇	—	2	1	no	—
232	C₂²×C₆₇	—	4	4	no	—
233	C₂²×C₆₇	—	4	8	no	—
234	C₂²×C₆₇	—	4	2	no	—
235	C₂²×C₆₇	—	4	2	no	—
236	C₂²×C₆₇	—	8	2	no	—
237	C₂²×C₆₇	—	8	2	no	—
238	C₂²×C₆₇	—	8	2	no	—
239	C₂²×C₆₇	—	16	4	no	—
240	C₂²×C₆₇	#472	1	2	yes	complemented
241	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
242	C₂²×C₆₇	#525	1	2	yes	—
243	C₄×C₆₇	—	2	1	no	—
244	C₄×C₆₇	—	2	4	no	—
245	C₄×C₆₇	—	2	1	no	—
246	C₄×C₆₇	—	8	2	no	—
247	C₄×C₆₇	C₂×#120	1	1	yes	characteristic
248	C₄×C₆₇	C₂×#121	1	2	yes	—
249	C₄×C₆₇	#525	1	1	yes	characteristic
250	#32942	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
251	C₂×C₆₇	—	2	2	no	—
252	C₂×C₆₇	—	2	2	no	—
253	C₂×C₆₇	—	4	1	no	—
254	C₂×C₆₇	—	4	4	no	—
255	C₂×C₆₇	—	16	2	no	—
256	C₂×C₆₇	#2847	1	1	yes	characteristic
257	C₂×C₆₇	C₂×#472	1	1	yes	characteristic
258	C₂×C₆₇	#3067	1	1	yes	characteristic
259	C₂²×#131	C₂×C₆₇	1	1	yes	characteristic, complemented
260	C₂×#472	C₂×C₆₇	1	2	yes	complemented
261	C₂×#466	C₂×C₆₇	1	1	yes	characteristic, complemented
262	#3006	C₂×C₆₇	1	2	yes	complemented
263	#1907	C₂×C₆₇	1	4	yes	complemented
264	#1900	C₂×C₆₇	1	2	yes	complemented
265	#1816	C₂×C₆₇	1	2	yes	complemented
266	#1810	C₂×C₆₇	1	1	yes	characteristic, complemented
267	C₆₇	#32942	1	1	yes	characteristic, direct product factor, Sylow
268	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
269	C₂×AGL(1,ℤ/8)	C₂²×C₆₇	1	4	yes	—
270	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
271	C₂×#131	—	2	8	no	—
272	C₂×#131	—	2	1	no	—
273	C₂×#131	—	2	1	no	—
274	C₂×#131	C₂²×C₆₇	1	2	yes	—
275	C₂×#131	C₂²×C₆₇	1	2	yes	—
276	C₂²×M₁₆	—	2	1	no	—
277	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
278	C₂³×C₈	—	2	1	no	—
279	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
280	C₂×#121	C₂²×C₆₇	1	2	yes	—
281	#472	—	2	8	no	—
282	#466	—	2	1	no	—
283	#466	—	2	1	no	—
284	#466	—	2	1	no	—
285	#466	—	2	1	no	—
286	#449	—	2	4	no	—
287	#448	—	2	2	no	—
288	#447	—	2	2	no	—
289	C₂×#108	C₂²×C₆₇	1	2	yes	complemented
290	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic, complemented
291	#417	C₂²×C₆₇	1	4	yes	—
292	C₂×#102	C₂²×C₆₇	1	2	yes	complemented
293	C₂×#100	C₂²×C₆₇	1	4	yes	complemented
294	#407	C₂²×C₆₇	1	2	yes	—
295	#343	C₂²×C₆₇	1	4	yes	complemented
296	#341	C₂²×C₆₇	1	2	yes	complemented
297	C₂×#41	—	2	4	no	—
298	C₂×#41	—	2	4	no	—
299	C₂×#41	—	2	1	no	—
300	C₂×#41	—	2	1	no	—
301	C₂×#41	D₄×C₆₇	1	2	yes	—
302	C₂×#41	C₂³×C₆₇	1	2	yes	—
303	C₂²×Q₈	D₄×C₆₇	1	1	yes	characteristic
304	C₂²×D₄	—	2	1	no	—
305	C₂²×D₄	—	2	2	no	—
306	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
307	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
308	C₂³×C₄	—	2	1	no	—
309	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
310	AGL(1,ℤ/8)	—	2	8	no	—
311	AGL(1,ℤ/8)	—	2	8	no	—
312	C₂×SD₁₆	—	2	4	no	—
313	C₂×D₈	—	2	4	no	—
314	C₂×D₈	—	2	1	no	—
315	C₂×D₈	—	2	1	no	—
316	C₂×D₈	—	2	4	no	—
317	#131	—	2	8	no	—
318	#131	—	2	8	no	—
319	#131	—	2	4	no	—
320	#131	—	2	4	no	—
321	#131	—	2	4	no	—
322	#131	D₄×C₆₇	1	8	yes	complemented
323	C₂×M₁₆	—	2	8	no	—
324	C₂×M₁₆	—	2	1	no	—
325	C₂×M₁₆	—	2	4	no	—
326	C₂×M₁₆	—	2	2	no	—
327	C₂×M₁₆	—	2	2	no	—
328	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
329	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
330	C₂²×C₈	—	2	8	no	—
331	C₂²×C₈	—	2	1	no	—
332	C₂²×C₈	—	2	2	no	—
333	C₂²×C₈	—	2	2	no	—
334	C₂²×C₈	—	2	4	no	—
335	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
336	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
337	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
338	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
339	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
340	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
341	#121	—	2	2	no	—
342	#121	—	2	2	no	—
343	#121	—	2	2	no	—
344	#121	—	2	2	no	—
345	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
346	C₂×C₂²⋊C₄	—	2	2	no	—
347	#108	—	2	4	no	—
348	#107	—	2	1	no	—
349	#107	—	2	1	no	—
350	#103	—	2	4	no	—
351	#102	—	2	4	no	—
352	#102	—	2	2	no	—
353	#102	—	2	2	no	—
354	#100	—	2	8	no	—
355	C₂²⋊C₈	—	2	2	no	—
356	C₂²⋊C₈	—	2	2	no	—
357	C₂⁴	—	2	1	no	—
358	C₂⁴	—	2	2	no	—
359	C₂⁴	#41×C₆₇	1	1	yes	characteristic
360	#41	—	2	4	no	—
361	#41	—	2	8	no	—
362	#41	—	2	4	no	—
363	#41	—	2	4	no	—
364	#41	—	2	4	no	—
365	#41	—	2	2	no	—
366	#41	—	2	2	no	—
367	#41	C₂×D₄×C₆₇	1	4	yes	—
368	#41	#41×C₆₇	1	4	yes	—
369	C₂×Q₈	—	2	4	no	—
370	C₂×Q₈	—	2	1	no	—
371	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
372	C₂×D₄	—	2	8	no	—
373	C₂×D₄	—	2	2	no	—
374	C₂×D₄	—	2	2	no	—
375	C₂×D₄	—	2	4	no	—
376	C₂×D₄	—	2	1	no	—
377	C₂×D₄	—	2	4	no	—
378	C₂×D₄	—	2	2	no	—
379	C₂×D₄	—	2	2	no	—
380	C₂×D₄	—	2	4	no	—
381	C₂×D₄	—	4	2	no	—
382	C₂×D₄	—	4	2	no	—
383	C₂×D₄	—	8	4	no	—
384	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
385	C₂²×C₄	—	2	4	no	—
386	C₂²×C₄	—	2	4	no	—
387	C₂²×C₄	—	2	1	no	—
388	C₂²×C₄	—	2	2	no	—
389	C₂²×C₄	—	2	2	no	—
390	C₂²×C₄	—	2	2	no	—
391	C₂²×C₄	—	2	2	no	—
392	C₂²×C₄	—	2	2	no	—
393	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
394	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
395	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
396	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
397	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
398	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
399	SD₁₆	—	4	8	no	—
400	D₈	—	4	8	no	—
401	D₈	—	4	2	no	—
402	D₈	—	4	2	no	—
403	M₁₆	—	2	4	no	—
404	M₁₆	—	2	8	no	—
405	M₁₆	—	2	2	no	—
406	M₁₆	—	2	8	no	—
407	M₁₆	—	2	4	no	—
408	M₁₆	—	2	4	no	—
409	M₁₆	C₂×D₄×C₆₇	1	4	yes	—
410	C₂×C₈	—	2	4	no	—
411	C₂×C₈	—	2	8	no	—
412	C₂×C₈	—	2	2	no	—
413	C₂×C₈	—	2	2	no	—
414	C₂×C₈	—	2	2	no	—
415	C₂×C₈	—	2	8	no	—
416	C₂×C₈	—	2	2	no	—
417	C₂×C₈	—	2	2	no	—
418	C₂×C₈	—	2	2	no	—
419	C₂×C₈	—	2	4	no	—
420	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
421	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
422	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
423	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
424	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
425	C₄⋊C₄	—	2	2	no	—
426	C₄⋊C₄	—	2	2	no	—
427	C₂²⋊C₄	—	4	2	no	—
428	C₂²⋊C₄	—	4	2	no	—
429	C₂³	—	2	4	no	—
430	C₂³	—	2	2	no	—
431	C₂³	—	2	2	no	—
432	C₂³	—	2	2	no	—
433	C₂³	—	4	1	no	—
434	C₂³	—	4	2	no	—
435	C₂³	—	4	2	no	—
436	C₂³	—	4	2	no	—
437	C₂³	—	4	4	no	—
438	C₂³	—	8	2	no	—
439	C₂³	—	8	2	no	—
440	C₂³	#121×C₆₇	1	1	yes	characteristic
441	C₂³	#121×C₆₇	1	1	yes	characteristic
442	C₂³	C₂×#41×C₆₇	1	1	yes	characteristic
443	Q₈	—	2	4	no	—
444	Q₈	—	2	2	no	—
445	Q₈	#121×C₆₇	1	4	yes	—
446	D₄	—	2	8	no	—
447	D₄	—	2	4	no	—
448	D₄	—	2	4	no	—
449	D₄	—	2	4	no	—
450	D₄	—	2	2	no	—
451	D₄	—	4	2	no	—
452	D₄	—	4	4	no	—
453	D₄	—	4	2	no	—
454	D₄	—	8	4	no	—
455	D₄	—	8	4	no	—
456	D₄	#121×C₆₇	1	4	yes	—
457	C₂×C₄	—	2	4	no	—
458	C₂×C₄	—	2	8	no	—
459	C₂×C₄	—	2	2	no	—
460	C₂×C₄	—	2	2	no	—
461	C₂×C₄	—	2	1	no	—
462	C₂×C₄	—	2	4	no	—
463	C₂×C₄	—	2	1	no	—
464	C₂×C₄	—	2	2	no	—
465	C₂×C₄	—	2	2	no	—
466	C₂×C₄	—	2	1	no	—
467	C₂×C₄	—	2	4	no	—
468	C₂×C₄	—	2	1	no	—
469	C₂×C₄	—	4	2	no	—
470	C₂×C₄	—	8	2	no	—
471	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
472	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
473	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
474	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
475	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
476	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
477	C₈	—	2	1	no	—
478	C₈	—	2	4	no	—
479	C₈	—	2	1	no	—
480	C₈	—	4	4	no	—
481	C₈	#120×C₆₇	1	2	yes	—
482	C₈	#120×C₆₇	1	2	yes	—
483	C₂²	—	2	1	no	—
484	C₂²	—	2	1	no	—
485	C₂²	—	2	1	no	—
486	C₂²	—	2	1	no	—
487	C₂²	—	2	1	no	—
488	C₂²	—	2	1	no	—
489	C₂²	—	2	1	no	—
490	C₂²	—	2	4	no	—
491	C₂²	—	4	4	no	—
492	C₂²	—	4	8	no	—
493	C₂²	—	4	2	no	—
494	C₂²	—	4	2	no	—
495	C₂²	—	8	2	no	—
496	C₂²	—	8	2	no	—
497	C₂²	—	8	2	no	—
498	C₂²	—	16	4	no	—
499	C₂²	#472×C₆₇	1	2	yes	complemented
500	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
501	C₂²	#525×C₆₇	1	2	yes	—
502	C₄	—	2	1	no	—
503	C₄	—	2	4	no	—
504	C₄	—	2	1	no	—
505	C₄	—	8	2	no	—
506	C₄	C₂×#120×C₆₇	1	1	yes	characteristic
507	C₄	C₂×#121×C₆₇	1	2	yes	—
508	C₄	#525×C₆₇	1	1	yes	characteristic
509	C₂	—	2	2	no	—
510	C₂	—	2	2	no	—
511	C₂	—	4	1	no	—
512	C₂	—	4	4	no	—
513	C₂	—	16	2	no	—
514	C₂	C₆₇×#2847	1	1	yes	characteristic
515	C₂	C₂×#472×C₆₇	1	1	yes	characteristic
516	C₂	C₆₇×#3067	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂³×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×#131×C₆₇	C₂	semidirect
2. C₂×#472×C₆₇	C₂	semidirect
3. C₂×#466×C₆₇	C₂	semidirect
4. C₆₇×#3006	C₂	semidirect
5. C₆₇×#1907	C₂	semidirect
6. C₆₇×#1900	C₂	semidirect
7. C₆₇×#1816	C₂	semidirect
8. C₆₇×#1810	C₂	semidirect
11. C₂²×D₈×C₆₇	C₂²	semidirect
30. C₂×#108×C₆₇	C₂²	semidirect
31. C₂×#107×C₆₇	C₂²	semidirect
33. C₂×#102×C₆₇	C₂²	semidirect
34. C₂×#100×C₆₇	C₂²	semidirect
36. #343×C₆₇	C₂²	semidirect
37. #341×C₆₇	C₂²	semidirect
63. #131×C₆₇	D₄	semidirect
240. C₂²×C₆₇	#472	semidirect
250. #32942	C₆₇	direct (N × H)
259. C₂²×#131	C₂×C₆₇	semidirect
260. C₂×#472	C₂×C₆₇	semidirect
261. C₂×#466	C₂×C₆₇	semidirect
262. #3006	C₂×C₆₇	semidirect
263. #1907	C₂×C₆₇	semidirect
264. #1900	C₂×C₆₇	semidirect
265. #1816	C₂×C₆₇	semidirect
266. #1810	C₂×C₆₇	semidirect
267. C₆₇	#32942	direct (N × H)
270. C₂²×D₈	C₂²×C₆₇	semidirect
289. C₂×#108	C₂²×C₆₇	semidirect
290. C₂×#107	C₂²×C₆₇	semidirect
292. C₂×#102	C₂²×C₆₇	semidirect
293. C₂×#100	C₂²×C₆₇	semidirect
295. #343	C₂²×C₆₇	semidirect
296. #341	C₂²×C₆₇	semidirect
322. #131	D₄×C₆₇	semidirect
499. C₂²	#472×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#131×C₆₇	2
C₂×#472×C₆₇	2
C₂×#466×C₆₇	2
C₆₇×#3006	2
C₆₇×#1907	2
C₆₇×#1900	2
C₆₇×#1816	2
C₆₇×#1810	2
#32942	67

Subgroups — C67×#32942 #30712193

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#32942 #30712193