Subgroups — C₆₇×#11157 #30690408

Order 17152.

922 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#418×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#418×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×#100×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#344×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#344×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1738	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×H(3,ℤ/4)×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
9	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
10	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
11	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
12	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
15	#418×C₆₇	—	2	2	no	—
16	#418×C₆₇	—	2	2	no	—
17	#418×C₆₇	—	2	2	no	—
18	#418×C₆₇	—	2	2	no	—
19	C₂×#102×C₆₇	—	2	2	no	—
20	C₂×#102×C₆₇	—	2	2	no	—
21	C₂×#100×C₆₇	—	2	2	no	—
22	C₂×#100×C₆₇	—	2	2	no	—
23	C₂×#100×C₆₇	C₄	1	4	yes	complemented
24	#344×C₆₇	—	2	2	no	—
25	#344×C₆₇	—	2	2	no	—
26	#341×C₆₇	—	2	2	no	—
27	H(3,ℤ/4)×C₆₇	—	2	2	no	—
28	C₂⁵×C₆₇	—	2	1	no	—
29	C₂²×D₄×C₆₇	—	2	1	no	—
30	C₂²×D₄×C₆₇	—	2	1	no	—
31	C₂²×D₄×C₆₇	—	2	1	no	—
32	C₂²×D₄×C₆₇	—	2	1	no	—
33	C₂²×D₄×C₆₇	—	2	1	no	—
34	C₂²×D₄×C₆₇	—	2	1	no	—
35	C₂²×D₄×C₆₇	—	2	1	no	—
36	C₂²×D₄×C₆₇	—	2	1	no	—
37	C₂²×D₄×C₆₇	—	2	1	no	—
38	C₂²×D₄×C₆₇	—	2	1	no	—
39	C₂²×D₄×C₆₇	—	2	1	no	—
40	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
41	C₂²×D₄×C₆₇	D₄	1	2	yes	—
42	C₂²×D₄×C₆₇	D₄	1	2	yes	—
43	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
44	C₂×M₁₆×C₆₇	—	2	1	no	—
45	C₂×M₁₆×C₆₇	—	2	1	no	—
46	C₂×M₁₆×C₆₇	—	2	1	no	—
47	C₂×M₁₆×C₆₇	—	2	1	no	—
48	C₂×M₁₆×C₆₇	—	2	1	no	—
49	C₂×M₁₆×C₆₇	—	2	1	no	—
50	C₂×M₁₆×C₆₇	—	2	1	no	—
51	C₂×M₁₆×C₆₇	—	2	1	no	—
52	C₂×M₁₆×C₆₇	—	2	1	no	—
53	C₂×M₁₆×C₆₇	—	2	1	no	—
54	C₂×M₁₆×C₆₇	—	2	1	no	—
55	C₂×M₁₆×C₆₇	—	2	1	no	—
56	C₂×M₁₆×C₆₇	—	2	1	no	—
57	C₂×M₁₆×C₆₇	—	2	1	no	—
58	C₂×M₁₆×C₆₇	—	2	1	no	—
59	C₂×M₁₆×C₆₇	—	2	1	no	—
60	C₂×M₁₆×C₆₇	—	2	1	no	—
61	C₂×M₁₆×C₆₇	—	2	1	no	—
62	C₂×M₁₆×C₆₇	—	2	1	no	—
63	C₂×M₁₆×C₆₇	—	2	1	no	—
64	C₂×M₁₆×C₆₇	—	2	1	no	—
65	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
66	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
67	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
68	C₂²×C₈×C₆₇	—	2	1	no	—
69	C₂²×C₈×C₆₇	—	2	1	no	—
70	C₂²×C₈×C₆₇	—	2	1	no	—
71	C₂²×C₈×C₆₇	D₄	1	2	yes	—
72	#117×C₆₇	—	2	1	no	—
73	#117×C₆₇	—	2	1	no	—
74	#117×C₆₇	—	2	1	no	—
75	#117×C₆₇	—	2	1	no	—
76	#117×C₆₇	—	2	1	no	—
77	#117×C₆₇	—	2	1	no	—
78	#117×C₆₇	D₄	1	2	yes	—
79	#117×C₆₇	D₄	1	2	yes	—
80	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
81	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
82	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
83	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
84	C₂×C₄²×C₆₇	—	2	1	no	—
85	C₂×C₄²×C₆₇	—	2	1	no	—
86	#102×C₆₇	—	2	8	no	—
87	#102×C₆₇	—	2	8	no	—
88	#100×C₆₇	—	2	8	no	—
89	C₂⁴×C₆₇	—	2	2	no	—
90	C₂⁴×C₆₇	—	2	2	no	—
91	C₂⁴×C₆₇	—	2	2	no	—
92	C₂⁴×C₆₇	—	4	1	no	—
93	C₂⁴×C₆₇	—	4	1	no	—
94	C₂⁴×C₆₇	—	4	1	no	—
95	C₂⁴×C₆₇	—	4	1	no	—
96	C₂⁴×C₆₇	—	8	4	no	—
97	C₂⁴×C₆₇	#41	1	1	yes	characteristic
98	C₂×D₄×C₆₇	—	2	2	no	—
99	C₂×D₄×C₆₇	—	2	2	no	—
100	C₂×D₄×C₆₇	—	2	2	no	—
101	C₂×D₄×C₆₇	—	2	2	no	—
102	C₂×D₄×C₆₇	—	2	2	no	—
103	C₂×D₄×C₆₇	—	2	2	no	—
104	C₂×D₄×C₆₇	—	2	2	no	—
105	C₂×D₄×C₆₇	—	2	2	no	—
106	C₂×D₄×C₆₇	—	2	2	no	—
107	C₂×D₄×C₆₇	—	2	2	no	—
108	C₂×D₄×C₆₇	—	2	2	no	—
109	C₂×D₄×C₆₇	—	2	2	no	—
110	C₂×D₄×C₆₇	—	2	2	no	—
111	C₂×D₄×C₆₇	—	2	2	no	—
112	C₂×D₄×C₆₇	—	2	2	no	—
113	C₂×D₄×C₆₇	—	4	1	no	—
114	C₂×D₄×C₆₇	—	4	1	no	—
115	C₂×D₄×C₆₇	—	4	1	no	—
116	C₂×D₄×C₆₇	—	4	1	no	—
117	C₂×D₄×C₆₇	—	4	1	no	—
118	C₂×D₄×C₆₇	—	4	1	no	—
119	C₂×D₄×C₆₇	—	4	1	no	—
120	C₂×D₄×C₆₇	—	4	1	no	—
121	C₂×D₄×C₆₇	—	4	1	no	—
122	C₂×D₄×C₆₇	—	4	1	no	—
123	C₂×D₄×C₆₇	—	4	1	no	—
124	C₂×D₄×C₆₇	—	4	1	no	—
125	C₂×D₄×C₆₇	C₂²⋊C₄	1	4	yes	—
126	C₂²×C₄×C₆₇	—	4	1	no	—
127	C₂²×C₄×C₆₇	—	4	1	no	—
128	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
129	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
130	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
131	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
132	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
133	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
134	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
135	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
136	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
137	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
138	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
139	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
140	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
141	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
142	M₁₆×C₆₇	—	2	2	no	—
143	M₁₆×C₆₇	—	2	2	no	—
144	M₁₆×C₆₇	—	2	2	no	—
145	M₁₆×C₆₇	—	2	2	no	—
146	M₁₆×C₆₇	—	2	2	no	—
147	M₁₆×C₆₇	—	2	2	no	—
148	M₁₆×C₆₇	—	4	1	no	—
149	M₁₆×C₆₇	—	4	1	no	—
150	M₁₆×C₆₇	—	4	1	no	—
151	M₁₆×C₆₇	—	4	1	no	—
152	M₁₆×C₆₇	—	4	1	no	—
153	M₁₆×C₆₇	—	4	1	no	—
154	M₁₆×C₆₇	—	4	1	no	—
155	M₁₆×C₆₇	—	4	1	no	—
156	M₁₆×C₆₇	—	4	1	no	—
157	M₁₆×C₆₇	—	4	1	no	—
158	C₂×C₈×C₆₇	—	2	2	no	—
159	C₂×C₈×C₆₇	—	2	2	no	—
160	C₂×C₈×C₆₇	—	2	2	no	—
161	C₂×C₈×C₆₇	—	2	2	no	—
162	C₂×C₈×C₆₇	—	2	2	no	—
163	C₂×C₈×C₆₇	—	2	2	no	—
164	C₂×C₈×C₆₇	—	4	1	no	—
165	C₂×C₈×C₆₇	—	4	1	no	—
166	C₂×C₈×C₆₇	—	4	1	no	—
167	C₂×C₈×C₆₇	—	4	1	no	—
168	C₂×C₈×C₆₇	—	4	1	no	—
169	C₂×C₈×C₆₇	—	4	1	no	—
170	C₄⋊C₄×C₆₇	—	2	2	no	—
171	C₄⋊C₄×C₆₇	—	2	2	no	—
172	C₄⋊C₄×C₆₇	—	4	1	no	—
173	C₄⋊C₄×C₆₇	—	4	1	no	—
174	C₂²⋊C₄×C₆₇	—	2	2	no	—
175	C₂²⋊C₄×C₆₇	—	2	2	no	—
176	C₂²⋊C₄×C₆₇	—	4	1	no	—
177	C₂²⋊C₄×C₆₇	—	4	1	no	—
178	C₄²×C₆₇	—	2	2	no	—
179	C₄²×C₆₇	—	2	2	no	—
180	C₄²×C₆₇	—	4	1	no	—
181	C₄²×C₆₇	—	4	1	no	—
182	C₂³×C₆₇	—	2	4	no	—
183	C₂³×C₆₇	—	4	2	no	—
184	C₂³×C₆₇	—	4	2	no	—
185	C₂³×C₆₇	—	4	2	no	—
186	C₂³×C₆₇	—	4	2	no	—
187	C₂³×C₆₇	—	4	2	no	—
188	C₂³×C₆₇	—	4	2	no	—
189	C₂³×C₆₇	—	4	2	no	—
190	C₂³×C₆₇	—	8	4	no	—
191	C₂³×C₆₇	—	8	4	no	—
192	C₂³×C₆₇	—	8	4	no	—
193	C₂³×C₆₇	—	8	4	no	—
194	C₂³×C₆₇	—	8	4	no	—
195	C₂³×C₆₇	—	8	4	no	—
196	C₂³×C₆₇	—	8	4	no	—
197	C₂³×C₆₇	#117	1	1	yes	characteristic
198	C₂³×C₆₇	#121	1	1	yes	characteristic
199	C₂³×C₆₇	#123	1	1	yes	characteristic
200	C₂³×C₆₇	#123	1	1	yes	characteristic
201	C₂³×C₆₇	#123	1	1	yes	characteristic
202	C₂³×C₆₇	#124	1	1	yes	characteristic
203	C₂³×C₆₇	#126	1	1	yes	characteristic
204	D₄×C₆₇	—	2	4	no	—
205	D₄×C₆₇	—	2	4	no	—
206	D₄×C₆₇	—	4	2	no	—
207	D₄×C₆₇	—	4	2	no	—
208	D₄×C₆₇	—	4	2	no	—
209	D₄×C₆₇	—	4	2	no	—
210	D₄×C₆₇	—	4	2	no	—
211	D₄×C₆₇	—	4	2	no	—
212	C₂×C₄×C₆₇	—	4	2	no	—
213	C₂×C₄×C₆₇	—	4	2	no	—
214	C₂×C₄×C₆₇	—	8	2	no	—
215	C₂×C₄×C₆₇	—	8	2	no	—
216	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
217	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
218	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
219	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
220	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
221	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
222	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
223	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
224	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
225	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
226	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
227	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
228	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
229	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
230	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
231	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
232	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
233	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
234	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
235	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
236	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
237	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
238	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
239	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
240	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
241	C₂×C₄×C₆₇	#126	1	1	yes	characteristic
242	C₂×C₄×C₆₇	#126	1	1	yes	characteristic
243	C₂×C₄×C₆₇	#126	1	1	yes	characteristic
244	C₈×C₆₇	—	4	2	no	—
245	C₈×C₆₇	—	4	2	no	—
246	C₈×C₆₇	—	4	2	no	—
247	C₈×C₆₇	—	4	2	no	—
248	C₂²×C₆₇	—	2	2	no	—
249	C₂²×C₆₇	—	2	2	no	—
250	C₂²×C₆₇	—	2	2	no	—
251	C₂²×C₆₇	—	4	1	no	—
252	C₂²×C₆₇	—	4	1	no	—
253	C₂²×C₆₇	—	4	1	no	—
254	C₂²×C₆₇	—	4	1	no	—
255	C₂²×C₆₇	—	4	4	no	—
256	C₂²×C₆₇	—	8	4	no	—
257	C₂²×C₆₇	—	8	4	no	—
258	C₂²×C₆₇	—	8	4	no	—
259	C₂²×C₆₇	—	8	4	no	—
260	C₂²×C₆₇	—	8	4	no	—
261	C₂²×C₆₇	—	8	4	no	—
262	C₂²×C₆₇	—	8	4	no	—
263	C₂²×C₆₇	#381	1	1	yes	characteristic
264	C₂²×C₆₇	#388	1	1	yes	characteristic
265	C₂²×C₆₇	#388	1	1	yes	characteristic
266	C₂²×C₆₇	#394	1	1	yes	characteristic
267	C₂²×C₆₇	#394	1	1	yes	characteristic
268	C₂²×C₆₇	#397	1	1	yes	characteristic
269	C₂²×C₆₇	#397	1	1	yes	characteristic
270	C₄×C₆₇	—	8	2	no	—
271	C₄×C₆₇	—	8	2	no	—
272	C₄×C₆₇	#386	1	1	yes	characteristic
273	C₄×C₆₇	#386	1	1	yes	characteristic
274	C₄×C₆₇	#388	1	1	yes	characteristic
275	C₄×C₆₇	#388	1	1	yes	characteristic
276	C₄×C₆₇	#394	1	1	yes	characteristic
277	C₄×C₆₇	#394	1	1	yes	characteristic
278	C₄×C₆₇	#397	1	1	yes	characteristic
279	C₄×C₆₇	#397	1	1	yes	characteristic
280	#11157	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
281	C₂×C₆₇	—	2	1	no	—
282	C₂×C₆₇	—	2	1	no	—
283	C₂×C₆₇	—	2	1	no	—
284	C₂×C₆₇	—	2	1	no	—
285	C₂×C₆₇	—	2	1	no	—
286	C₂×C₆₇	—	2	1	no	—
287	C₂×C₆₇	—	8	4	no	—
288	C₂×C₆₇	#1440	1	1	yes	characteristic
289	C₂×C₆₇	#2081	1	2	yes	—
290	C₂×#418	C₂×C₆₇	1	1	yes	characteristic
291	C₂×#418	C₂×C₆₇	1	1	yes	characteristic
292	C₂²×#100	C₂×C₆₇	1	1	yes	characteristic
293	C₂×#344	C₂×C₆₇	1	1	yes	characteristic, complemented
294	C₂×#344	C₂×C₆₇	1	1	yes	characteristic, complemented
295	#1738	C₂×C₆₇	1	1	yes	characteristic, complemented
296	C₂×H(3,ℤ/4)	C₂×C₆₇	1	1	yes	characteristic, complemented
297	C₆₇	#11157	1	1	yes	characteristic, direct product factor, Sylow
298	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
299	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
300	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
301	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
302	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
303	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
304	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
305	#418	—	2	2	no	—
306	#418	—	2	2	no	—
307	#418	—	2	2	no	—
308	#418	—	2	2	no	—
309	C₂×#102	—	2	2	no	—
310	C₂×#102	—	2	2	no	—
311	C₂×#100	—	2	2	no	—
312	C₂×#100	—	2	2	no	—
313	C₂×#100	C₄×C₆₇	1	4	yes	complemented
314	#344	—	2	2	no	—
315	#344	—	2	2	no	—
316	#341	—	2	2	no	—
317	H(3,ℤ/4)	—	2	2	no	—
318	C₂⁵	—	2	1	no	—
319	C₂²×D₄	—	2	1	no	—
320	C₂²×D₄	—	2	1	no	—
321	C₂²×D₄	—	2	1	no	—
322	C₂²×D₄	—	2	1	no	—
323	C₂²×D₄	—	2	1	no	—
324	C₂²×D₄	—	2	1	no	—
325	C₂²×D₄	—	2	1	no	—
326	C₂²×D₄	—	2	1	no	—
327	C₂²×D₄	—	2	1	no	—
328	C₂²×D₄	—	2	1	no	—
329	C₂²×D₄	—	2	1	no	—
330	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	—
331	C₂²×D₄	D₄×C₆₇	1	2	yes	—
332	C₂²×D₄	D₄×C₆₇	1	2	yes	—
333	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
334	C₂×M₁₆	—	2	1	no	—
335	C₂×M₁₆	—	2	1	no	—
336	C₂×M₁₆	—	2	1	no	—
337	C₂×M₁₆	—	2	1	no	—
338	C₂×M₁₆	—	2	1	no	—
339	C₂×M₁₆	—	2	1	no	—
340	C₂×M₁₆	—	2	1	no	—
341	C₂×M₁₆	—	2	1	no	—
342	C₂×M₁₆	—	2	1	no	—
343	C₂×M₁₆	—	2	1	no	—
344	C₂×M₁₆	—	2	1	no	—
345	C₂×M₁₆	—	2	1	no	—
346	C₂×M₁₆	—	2	1	no	—
347	C₂×M₁₆	—	2	1	no	—
348	C₂×M₁₆	—	2	1	no	—
349	C₂×M₁₆	—	2	1	no	—
350	C₂×M₁₆	—	2	1	no	—
351	C₂×M₁₆	—	2	1	no	—
352	C₂×M₁₆	—	2	1	no	—
353	C₂×M₁₆	—	2	1	no	—
354	C₂×M₁₆	—	2	1	no	—
355	C₂×M₁₆	C₂×C₄×C₆₇	1	2	yes	—
356	C₂×M₁₆	C₂×C₄×C₆₇	1	2	yes	—
357	C₂×M₁₆	D₄×C₆₇	1	2	yes	—
358	C₂²×C₈	—	2	1	no	—
359	C₂²×C₈	—	2	1	no	—
360	C₂²×C₈	—	2	1	no	—
361	C₂²×C₈	D₄×C₆₇	1	2	yes	—
362	#117	—	2	1	no	—
363	#117	—	2	1	no	—
364	#117	—	2	1	no	—
365	#117	—	2	1	no	—
366	#117	—	2	1	no	—
367	#117	—	2	1	no	—
368	#117	D₄×C₆₇	1	2	yes	—
369	#117	D₄×C₆₇	1	2	yes	—
370	C₂×C₄⋊C₄	—	2	1	no	—
371	C₂×C₄⋊C₄	—	2	1	no	—
372	C₂×C₂²⋊C₄	—	2	1	no	—
373	C₂×C₂²⋊C₄	—	2	1	no	—
374	C₂×C₄²	—	2	1	no	—
375	C₂×C₄²	—	2	1	no	—
376	#102	—	2	8	no	—
377	#102	—	2	8	no	—
378	#100	—	2	8	no	—
379	C₂⁴	—	2	2	no	—
380	C₂⁴	—	2	2	no	—
381	C₂⁴	—	2	2	no	—
382	C₂⁴	—	4	1	no	—
383	C₂⁴	—	4	1	no	—
384	C₂⁴	—	4	1	no	—
385	C₂⁴	—	4	1	no	—
386	C₂⁴	—	8	4	no	—
387	C₂⁴	#41×C₆₇	1	1	yes	characteristic
388	C₂×D₄	—	2	2	no	—
389	C₂×D₄	—	2	2	no	—
390	C₂×D₄	—	2	2	no	—
391	C₂×D₄	—	2	2	no	—
392	C₂×D₄	—	2	2	no	—
393	C₂×D₄	—	2	2	no	—
394	C₂×D₄	—	2	2	no	—
395	C₂×D₄	—	2	2	no	—
396	C₂×D₄	—	2	2	no	—
397	C₂×D₄	—	2	2	no	—
398	C₂×D₄	—	2	2	no	—
399	C₂×D₄	—	2	2	no	—
400	C₂×D₄	—	2	2	no	—
401	C₂×D₄	—	2	2	no	—
402	C₂×D₄	—	2	2	no	—
403	C₂×D₄	—	4	1	no	—
404	C₂×D₄	—	4	1	no	—
405	C₂×D₄	—	4	1	no	—
406	C₂×D₄	—	4	1	no	—
407	C₂×D₄	—	4	1	no	—
408	C₂×D₄	—	4	1	no	—
409	C₂×D₄	—	4	1	no	—
410	C₂×D₄	—	4	1	no	—
411	C₂×D₄	—	4	1	no	—
412	C₂×D₄	—	4	1	no	—
413	C₂×D₄	—	4	1	no	—
414	C₂×D₄	—	4	1	no	—
415	C₂×D₄	C₂²⋊C₄×C₆₇	1	4	yes	—
416	C₂²×C₄	—	4	1	no	—
417	C₂²×C₄	—	4	1	no	—
418	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
419	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
420	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
421	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
422	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
423	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
424	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
425	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
426	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
427	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
428	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
429	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
430	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
431	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
432	M₁₆	—	2	2	no	—
433	M₁₆	—	2	2	no	—
434	M₁₆	—	2	2	no	—
435	M₁₆	—	2	2	no	—
436	M₁₆	—	2	2	no	—
437	M₁₆	—	2	2	no	—
438	M₁₆	—	4	1	no	—
439	M₁₆	—	4	1	no	—
440	M₁₆	—	4	1	no	—
441	M₁₆	—	4	1	no	—
442	M₁₆	—	4	1	no	—
443	M₁₆	—	4	1	no	—
444	M₁₆	—	4	1	no	—
445	M₁₆	—	4	1	no	—
446	M₁₆	—	4	1	no	—
447	M₁₆	—	4	1	no	—
448	C₂×C₈	—	2	2	no	—
449	C₂×C₈	—	2	2	no	—
450	C₂×C₈	—	2	2	no	—
451	C₂×C₈	—	2	2	no	—
452	C₂×C₈	—	2	2	no	—
453	C₂×C₈	—	2	2	no	—
454	C₂×C₈	—	4	1	no	—
455	C₂×C₈	—	4	1	no	—
456	C₂×C₈	—	4	1	no	—
457	C₂×C₈	—	4	1	no	—
458	C₂×C₈	—	4	1	no	—
459	C₂×C₈	—	4	1	no	—
460	C₄⋊C₄	—	2	2	no	—
461	C₄⋊C₄	—	2	2	no	—
462	C₄⋊C₄	—	4	1	no	—
463	C₄⋊C₄	—	4	1	no	—
464	C₂²⋊C₄	—	2	2	no	—
465	C₂²⋊C₄	—	2	2	no	—
466	C₂²⋊C₄	—	4	1	no	—
467	C₂²⋊C₄	—	4	1	no	—
468	C₄²	—	2	2	no	—
469	C₄²	—	2	2	no	—
470	C₄²	—	4	1	no	—
471	C₄²	—	4	1	no	—
472	C₂³	—	2	4	no	—
473	C₂³	—	4	2	no	—
474	C₂³	—	4	2	no	—
475	C₂³	—	4	2	no	—
476	C₂³	—	4	2	no	—
477	C₂³	—	4	2	no	—
478	C₂³	—	4	2	no	—
479	C₂³	—	4	2	no	—
480	C₂³	—	8	4	no	—
481	C₂³	—	8	4	no	—
482	C₂³	—	8	4	no	—
483	C₂³	—	8	4	no	—
484	C₂³	—	8	4	no	—
485	C₂³	—	8	4	no	—
486	C₂³	—	8	4	no	—
487	C₂³	#117×C₆₇	1	1	yes	characteristic
488	C₂³	#121×C₆₇	1	1	yes	characteristic
489	C₂³	#123×C₆₇	1	1	yes	characteristic
490	C₂³	#123×C₆₇	1	1	yes	characteristic
491	C₂³	#123×C₆₇	1	1	yes	characteristic
492	C₂³	#124×C₆₇	1	1	yes	characteristic
493	C₂³	#126×C₆₇	1	1	yes	characteristic
494	D₄	—	2	4	no	—
495	D₄	—	2	4	no	—
496	D₄	—	4	2	no	—
497	D₄	—	4	2	no	—
498	D₄	—	4	2	no	—
499	D₄	—	4	2	no	—
500	D₄	—	4	2	no	—
501	D₄	—	4	2	no	—
502	C₂×C₄	—	4	2	no	—
503	C₂×C₄	—	4	2	no	—
504	C₂×C₄	—	8	2	no	—
505	C₂×C₄	—	8	2	no	—
506	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
507	C₂×C₄	#117×C₆₇	1	1	yes	characteristic
508	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
509	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
510	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
511	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
512	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
513	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
514	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
515	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
516	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
517	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
518	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
519	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
520	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
521	C₂×C₄	#123×C₆₇	1	1	yes	characteristic
522	C₂×C₄	#123×C₆₇	1	1	yes	characteristic
523	C₂×C₄	#123×C₆₇	1	1	yes	characteristic
524	C₂×C₄	#123×C₆₇	1	1	yes	characteristic
525	C₂×C₄	#123×C₆₇	1	1	yes	characteristic
526	C₂×C₄	#123×C₆₇	1	1	yes	characteristic
527	C₂×C₄	#123×C₆₇	1	1	yes	characteristic
528	C₂×C₄	#124×C₆₇	1	1	yes	characteristic
529	C₂×C₄	#124×C₆₇	1	1	yes	characteristic
530	C₂×C₄	#124×C₆₇	1	1	yes	characteristic
531	C₂×C₄	#126×C₆₇	1	1	yes	characteristic
532	C₂×C₄	#126×C₆₇	1	1	yes	characteristic
533	C₂×C₄	#126×C₆₇	1	1	yes	characteristic
534	C₈	—	4	2	no	—
535	C₈	—	4	2	no	—
536	C₈	—	4	2	no	—
537	C₈	—	4	2	no	—
538	C₂²	—	2	2	no	—
539	C₂²	—	2	2	no	—
540	C₂²	—	2	2	no	—
541	C₂²	—	4	1	no	—
542	C₂²	—	4	1	no	—
543	C₂²	—	4	1	no	—
544	C₂²	—	4	1	no	—
545	C₂²	—	4	4	no	—
546	C₂²	—	8	4	no	—
547	C₂²	—	8	4	no	—
548	C₂²	—	8	4	no	—
549	C₂²	—	8	4	no	—
550	C₂²	—	8	4	no	—
551	C₂²	—	8	4	no	—
552	C₂²	—	8	4	no	—
553	C₂²	#381×C₆₇	1	1	yes	characteristic
554	C₂²	#388×C₆₇	1	1	yes	characteristic
555	C₂²	#388×C₆₇	1	1	yes	characteristic
556	C₂²	#394×C₆₇	1	1	yes	characteristic
557	C₂²	#394×C₆₇	1	1	yes	characteristic
558	C₂²	#397×C₆₇	1	1	yes	characteristic
559	C₂²	#397×C₆₇	1	1	yes	characteristic
560	C₄	—	8	2	no	—
561	C₄	—	8	2	no	—
562	C₄	#386×C₆₇	1	1	yes	characteristic
563	C₄	#386×C₆₇	1	1	yes	characteristic
564	C₄	#388×C₆₇	1	1	yes	characteristic
565	C₄	#388×C₆₇	1	1	yes	characteristic
566	C₄	#394×C₆₇	1	1	yes	characteristic
567	C₄	#394×C₆₇	1	1	yes	characteristic
568	C₄	#397×C₆₇	1	1	yes	characteristic
569	C₄	#397×C₆₇	1	1	yes	characteristic
570	C₂	—	2	1	no	—
571	C₂	—	2	1	no	—
572	C₂	—	2	1	no	—
573	C₂	—	2	1	no	—
574	C₂	—	2	1	no	—
575	C₂	—	2	1	no	—
576	C₂	—	8	4	no	—
577	C₂	C₆₇×#1440	1	1	yes	characteristic
578	C₂	C₆₇×#2081	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₇	#397
commutator	C₂²×C₄	C₂²×C₄×C₆₇
Frattini, cosocle	C₂³×C₄	C₂³×C₆₇
prime radical	C₂³×D₄×C₆₇	C₂²
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
4. C₂×#344×C₆₇	C₂	semidirect
5. C₂×#344×C₆₇	C₂	semidirect
6. C₆₇×#1738	C₂	semidirect
7. C₂×H(3,ℤ/4)×C₆₇	C₂	semidirect
23. C₂×#100×C₆₇	C₄	semidirect
280. #11157	C₆₇	direct (N × H)
293. C₂×#344	C₂×C₆₇	semidirect
294. C₂×#344	C₂×C₆₇	semidirect
295. #1738	C₂×C₆₇	semidirect
296. C₂×H(3,ℤ/4)	C₂×C₆₇	semidirect
297. C₆₇	#11157	direct (N × H)
313. C₂×#100	C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#418×C₆₇	2
C₂²×#100×C₆₇	2
C₂×#344×C₆₇	2
C₆₇×#1738	2
C₂×H(3,ℤ/4)×C₆₇	2
#11157	67

Subgroups — C67×#11157 #30690408

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#11157 #30690408