Subgroups — C₆₇×#21091 #30700342

Order 17152.

1234 conjugacy classes of subgroups.

The subgroup lattice has 468 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	C₂²×#102×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#405×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2317	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#2309	C₂	1	1	yes	maximal, characteristic
6	C₆₇×#1936	C₂	1	4	yes	maximal, complemented
7	C₆₇×#1934	C₂	1	4	yes	maximal, complemented
8	C₆₇×#1736	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1735	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
11	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
12	C₂³×C₈×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	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
16	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
17	#418×C₆₇	C₄	1	8	yes	complemented
18	C₂×#102×C₆₇	—	2	4	no	—
19	C₂×#102×C₆₇	C₂²	1	2	yes	—
20	C₂×#102×C₆₇	C₂²	1	2	yes	—
21	C₂×#102×C₆₇	C₂²	1	4	yes	—
22	#405×C₆₇	—	2	4	no	—
23	C₂×#97×C₆₇	C₂²	1	2	yes	complemented
24	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	complemented
25	#390×C₆₇	—	2	1	no	—
26	#390×C₆₇	—	2	1	no	—
27	#390×C₆₇	C₂²	1	4	yes	—
28	#389×C₆₇	C₂²	1	4	yes	complemented
29	#385×C₆₇	—	2	2	no	—
30	#384×C₆₇	—	2	1	no	—
31	#382×C₆₇	—	2	1	no	—
32	#379×C₆₇	—	2	2	no	—
33	#340×C₆₇	C₂²	1	2	yes	complemented
34	#340×C₆₇	C₂²	1	2	yes	complemented
35	#340×C₆₇	C₂²	1	4	yes	complemented
36	C₂⁵×C₆₇	—	2	1	no	—
37	C₂²×D₄×C₆₇	—	2	2	no	—
38	C₂²×D₄×C₆₇	—	2	1	no	—
39	C₂²×D₄×C₆₇	—	2	2	no	—
40	C₂²×D₄×C₆₇	—	2	2	no	—
41	C₂²×D₄×C₆₇	—	2	2	no	—
42	C₂²×D₄×C₆₇	—	2	2	no	—
43	C₂²×D₄×C₆₇	C₂×C₄	1	4	yes	—
44	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
45	C₂³×C₄×C₆₇	—	2	1	no	—
46	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
47	C₂×M₁₆×C₆₇	—	2	2	no	—
48	C₂×M₁₆×C₆₇	—	2	2	no	—
49	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
50	C₂²×C₈×C₆₇	—	2	2	no	—
51	C₂²×C₈×C₆₇	—	2	2	no	—
52	C₂²×C₈×C₆₇	D₄	1	4	yes	—
53	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
54	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
55	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
56	#117×C₆₇	—	2	2	no	—
57	#117×C₆₇	—	2	2	no	—
58	#117×C₆₇	—	2	2	no	—
59	#117×C₆₇	—	2	2	no	—
60	#117×C₆₇	—	2	2	no	—
61	#117×C₆₇	C₂×C₄	1	4	yes	—
62	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
63	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
64	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
65	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
66	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
67	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
68	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
69	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
70	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
71	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
72	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
73	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
74	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
75	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
76	C₂×C₄²×C₆₇	—	2	1	no	—
77	C₂×C₄²×C₆₇	—	2	1	no	—
78	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
79	#102×C₆₇	—	2	4	no	—
80	#102×C₆₇	—	2	4	no	—
81	#102×C₆₇	C₂×C₄	1	16	yes	complemented
82	#97×C₆₇	—	2	4	no	—
83	C₄×C₈×C₆₇	—	2	4	no	—
84	#95×C₆₇	—	2	4	no	—
85	C₂⁴×C₆₇	—	2	2	no	—
86	C₂⁴×C₆₇	—	2	4	no	—
87	C₂⁴×C₆₇	—	4	2	no	—
88	C₂⁴×C₆₇	—	4	2	no	—
89	C₂⁴×C₆₇	—	8	2	no	—
90	C₂⁴×C₆₇	—	8	2	no	—
91	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
92	C₂×D₄×C₆₇	—	2	8	no	—
93	C₂×D₄×C₆₇	—	2	8	no	—
94	C₂×D₄×C₆₇	—	2	4	no	—
95	C₂×D₄×C₆₇	—	2	2	no	—
96	C₂×D₄×C₆₇	—	2	2	no	—
97	C₂×D₄×C₆₇	—	2	2	no	—
98	C₂×D₄×C₆₇	—	2	2	no	—
99	C₂×D₄×C₆₇	—	2	2	no	—
100	C₂×D₄×C₆₇	—	4	2	no	—
101	C₂×D₄×C₆₇	—	4	4	no	—
102	C₂×D₄×C₆₇	—	4	2	no	—
103	C₂×D₄×C₆₇	—	4	4	no	—
104	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
105	C₂²×C₄×C₆₇	—	2	2	no	—
106	C₂²×C₄×C₆₇	—	2	2	no	—
107	C₂²×C₄×C₆₇	—	2	2	no	—
108	C₂²×C₄×C₆₇	—	4	2	no	—
109	C₂²×C₄×C₆₇	—	8	2	no	—
110	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
111	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
112	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
113	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
114	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
115	C₂²×C₄×C₆₇	#41	1	2	yes	—
116	C₂²×C₄×C₆₇	#41	1	2	yes	—
117	C₂²×C₄×C₆₇	#41	1	2	yes	—
118	C₂²×C₄×C₆₇	#41	1	2	yes	—
119	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
120	M₁₆×C₆₇	—	2	8	no	—
121	C₂×C₈×C₆₇	—	2	8	no	—
122	C₂×C₈×C₆₇	—	2	2	no	—
123	C₂×C₈×C₆₇	—	2	1	no	—
124	C₂×C₈×C₆₇	—	2	1	no	—
125	C₂×C₈×C₆₇	—	2	1	no	—
126	C₂×C₈×C₆₇	—	2	1	no	—
127	C₂×C₈×C₆₇	—	2	2	no	—
128	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
129	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
130	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
131	C₄⋊C₄×C₆₇	—	2	2	no	—
132	C₄⋊C₄×C₆₇	—	2	2	no	—
133	C₄⋊C₄×C₆₇	—	2	2	no	—
134	C₄⋊C₄×C₆₇	—	2	2	no	—
135	C₄⋊C₄×C₆₇	—	2	2	no	—
136	C₄⋊C₄×C₆₇	—	2	2	no	—
137	C₄⋊C₄×C₆₇	—	2	2	no	—
138	C₄⋊C₄×C₆₇	—	4	2	no	—
139	C₄⋊C₄×C₆₇	—	8	4	no	—
140	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
141	C₂²⋊C₄×C₆₇	—	2	4	no	—
142	C₂²⋊C₄×C₆₇	—	2	4	no	—
143	C₂²⋊C₄×C₆₇	—	4	2	no	—
144	C₂²⋊C₄×C₆₇	—	8	4	no	—
145	C₂²⋊C₄×C₆₇	—	8	4	no	—
146	C₄²×C₆₇	—	2	4	no	—
147	C₄²×C₆₇	—	2	2	no	—
148	C₄²×C₆₇	—	2	2	no	—
149	C₄²×C₆₇	—	4	2	no	—
150	C₂³×C₆₇	—	2	2	no	—
151	C₂³×C₆₇	—	2	2	no	—
152	C₂³×C₆₇	—	2	4	no	—
153	C₂³×C₆₇	—	4	2	no	—
154	C₂³×C₆₇	—	4	2	no	—
155	C₂³×C₆₇	—	4	4	no	—
156	C₂³×C₆₇	—	4	4	no	—
157	C₂³×C₆₇	—	4	8	no	—
158	C₂³×C₆₇	—	8	8	no	—
159	C₂³×C₆₇	—	8	8	no	—
160	C₂³×C₆₇	—	8	4	no	—
161	C₂³×C₆₇	—	8	4	no	—
162	C₂³×C₆₇	C₄×D₄	1	2	yes	—
163	C₂³×C₆₇	#121	1	2	yes	—
164	C₂³×C₆₇	#124	1	2	yes	—
165	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
166	D₄×C₆₇	—	2	4	no	—
167	D₄×C₆₇	—	2	4	no	—
168	D₄×C₆₇	—	4	2	no	—
169	D₄×C₆₇	—	4	2	no	—
170	D₄×C₆₇	—	4	8	no	—
171	C₂×C₄×C₆₇	—	2	4	no	—
172	C₂×C₄×C₆₇	—	2	2	no	—
173	C₂×C₄×C₆₇	—	2	2	no	—
174	C₂×C₄×C₆₇	—	2	4	no	—
175	C₂×C₄×C₆₇	—	2	4	no	—
176	C₂×C₄×C₆₇	—	4	2	no	—
177	C₂×C₄×C₆₇	—	4	2	no	—
178	C₂×C₄×C₆₇	—	4	2	no	—
179	C₂×C₄×C₆₇	—	4	2	no	—
180	C₂×C₄×C₆₇	—	4	2	no	—
181	C₂×C₄×C₆₇	—	4	4	no	—
182	C₂×C₄×C₆₇	—	4	2	no	—
183	C₂×C₄×C₆₇	—	8	4	no	—
184	C₂×C₄×C₆₇	#117	1	4	yes	—
185	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
186	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
187	C₂×C₄×C₆₇	#121	1	2	yes	—
188	C₂×C₄×C₆₇	#123	1	4	yes	—
189	C₂×C₄×C₆₇	#124	1	2	yes	—
190	C₂×C₄×C₆₇	#126	1	4	yes	—
191	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
192	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
193	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
194	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
195	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
196	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
197	C₈×C₆₇	—	2	4	no	—
198	C₈×C₆₇	—	2	2	no	—
199	C₈×C₆₇	—	2	2	no	—
200	C₂²×C₆₇	—	2	4	no	—
201	C₂²×C₆₇	—	2	4	no	—
202	C₂²×C₆₇	—	2	2	no	—
203	C₂²×C₆₇	—	2	2	no	—
204	C₂²×C₆₇	—	4	8	no	—
205	C₂²×C₆₇	—	4	4	no	—
206	C₂²×C₆₇	—	8	16	no	—
207	C₂²×C₆₇	—	8	4	no	—
208	C₂²×C₆₇	—	8	2	no	—
209	C₂²×C₆₇	—	8	2	no	—
210	C₂²×C₆₇	#388	1	4	yes	—
211	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
212	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
213	C₂²×C₆₇	C₂×#124	1	1	yes	characteristic
214	C₂²×C₆₇	#576	1	2	yes	—
215	C₂²×C₆₇	#577	1	2	yes	—
216	C₄×C₆₇	—	4	4	no	—
217	C₄×C₆₇	—	4	2	no	—
218	C₄×C₆₇	—	4	2	no	—
219	C₄×C₆₇	—	8	2	no	—
220	C₄×C₆₇	—	8	2	no	—
221	C₄×C₆₇	#388	1	4	yes	—
222	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
223	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
224	C₄×C₆₇	C₂×#123	1	1	yes	characteristic
225	C₄×C₆₇	C₂×#126	1	1	yes	characteristic
226	#21091	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
227	C₂×C₆₇	—	2	4	no	—
228	C₂×C₆₇	—	8	4	no	—
229	C₂×C₆₇	C₂×#388	1	1	yes	characteristic
230	C₂×C₆₇	#2946	1	2	yes	—
231	C₂×C₆₇	#3057	1	1	yes	characteristic
232	C₂×C₆₇	#3058	1	1	yes	characteristic
233	C₂×C₆₇	#3136	1	2	yes	—
234	C₂×#418	C₂×C₆₇	1	1	yes	characteristic
235	C₂²×#102	C₂×C₆₇	1	1	yes	characteristic
236	C₂×#405	C₂×C₆₇	1	1	yes	characteristic, complemented
237	#2317	C₂×C₆₇	1	1	yes	characteristic, complemented
238	#2309	C₂×C₆₇	1	1	yes	characteristic
239	#1936	C₂×C₆₇	1	4	yes	complemented
240	#1934	C₂×C₆₇	1	4	yes	complemented
241	#1736	C₂×C₆₇	1	1	yes	characteristic, complemented
242	#1735	C₂×C₆₇	1	1	yes	characteristic, complemented
243	C₆₇	#21091	1	1	yes	characteristic, direct product factor, Sylow
244	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
245	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
246	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
247	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
248	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
249	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
250	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
251	#418	C₄×C₆₇	1	8	yes	complemented
252	C₂×#102	—	2	4	no	—
253	C₂×#102	C₂²×C₆₇	1	2	yes	—
254	C₂×#102	C₂²×C₆₇	1	2	yes	—
255	C₂×#102	C₂²×C₆₇	1	4	yes	—
256	#405	—	2	4	no	—
257	C₂×#97	C₂²×C₆₇	1	2	yes	complemented
258	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	complemented
259	#390	—	2	1	no	—
260	#390	—	2	1	no	—
261	#390	C₂²×C₆₇	1	4	yes	—
262	#389	C₂²×C₆₇	1	4	yes	complemented
263	#385	—	2	2	no	—
264	#384	—	2	1	no	—
265	#382	—	2	1	no	—
266	#379	—	2	2	no	—
267	#340	C₂²×C₆₇	1	2	yes	complemented
268	#340	C₂²×C₆₇	1	4	yes	complemented
269	#340	C₂²×C₆₇	1	2	yes	complemented
270	C₂⁵	—	2	1	no	—
271	C₂²×D₄	—	2	2	no	—
272	C₂²×D₄	—	2	1	no	—
273	C₂²×D₄	—	2	2	no	—
274	C₂²×D₄	—	2	2	no	—
275	C₂²×D₄	—	2	2	no	—
276	C₂²×D₄	—	2	2	no	—
277	C₂²×D₄	C₂×C₄×C₆₇	1	4	yes	—
278	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
279	C₂³×C₄	—	2	1	no	—
280	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
281	C₂×M₁₆	—	2	2	no	—
282	C₂×M₁₆	—	2	2	no	—
283	C₂×M₁₆	C₂×C₄×C₆₇	1	4	yes	—
284	C₂²×C₈	—	2	2	no	—
285	C₂²×C₈	—	2	2	no	—
286	C₂²×C₈	D₄×C₆₇	1	4	yes	—
287	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
288	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
289	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
290	#117	—	2	2	no	—
291	#117	—	2	2	no	—
292	#117	—	2	2	no	—
293	#117	—	2	2	no	—
294	#117	—	2	2	no	—
295	#117	C₂×C₄×C₆₇	1	4	yes	—
296	C₂×C₄⋊C₄	—	2	2	no	—
297	C₂×C₄⋊C₄	—	2	2	no	—
298	C₂×C₄⋊C₄	—	2	1	no	—
299	C₂×C₄⋊C₄	—	2	4	no	—
300	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
301	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
302	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
303	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
304	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
305	C₂×C₂²⋊C₄	—	2	1	no	—
306	C₂×C₂²⋊C₄	—	2	1	no	—
307	C₂×C₂²⋊C₄	—	2	4	no	—
308	C₂×C₂²⋊C₄	—	2	4	no	—
309	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
310	C₂×C₄²	—	2	1	no	—
311	C₂×C₄²	—	2	1	no	—
312	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
313	#102	—	2	4	no	—
314	#102	—	2	4	no	—
315	#102	C₂×C₄×C₆₇	1	16	yes	complemented
316	#97	—	2	4	no	—
317	C₄×C₈	—	2	4	no	—
318	#95	—	2	4	no	—
319	C₂⁴	—	2	2	no	—
320	C₂⁴	—	2	4	no	—
321	C₂⁴	—	4	2	no	—
322	C₂⁴	—	4	2	no	—
323	C₂⁴	—	8	2	no	—
324	C₂⁴	—	8	2	no	—
325	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
326	C₂×D₄	—	2	8	no	—
327	C₂×D₄	—	2	8	no	—
328	C₂×D₄	—	2	4	no	—
329	C₂×D₄	—	2	2	no	—
330	C₂×D₄	—	2	2	no	—
331	C₂×D₄	—	2	2	no	—
332	C₂×D₄	—	2	2	no	—
333	C₂×D₄	—	2	2	no	—
334	C₂×D₄	—	4	2	no	—
335	C₂×D₄	—	4	4	no	—
336	C₂×D₄	—	4	2	no	—
337	C₂×D₄	—	4	4	no	—
338	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
339	C₂²×C₄	—	2	2	no	—
340	C₂²×C₄	—	2	2	no	—
341	C₂²×C₄	—	2	2	no	—
342	C₂²×C₄	—	4	2	no	—
343	C₂²×C₄	—	8	2	no	—
344	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
345	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
346	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
347	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
348	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
349	C₂²×C₄	#41×C₆₇	1	2	yes	—
350	C₂²×C₄	#41×C₆₇	1	2	yes	—
351	C₂²×C₄	#41×C₆₇	1	2	yes	—
352	C₂²×C₄	#41×C₆₇	1	2	yes	—
353	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
354	M₁₆	—	2	8	no	—
355	C₂×C₈	—	2	8	no	—
356	C₂×C₈	—	2	2	no	—
357	C₂×C₈	—	2	1	no	—
358	C₂×C₈	—	2	1	no	—
359	C₂×C₈	—	2	1	no	—
360	C₂×C₈	—	2	1	no	—
361	C₂×C₈	—	2	2	no	—
362	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
363	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
364	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
365	C₄⋊C₄	—	2	2	no	—
366	C₄⋊C₄	—	2	2	no	—
367	C₄⋊C₄	—	2	2	no	—
368	C₄⋊C₄	—	2	2	no	—
369	C₄⋊C₄	—	2	2	no	—
370	C₄⋊C₄	—	2	2	no	—
371	C₄⋊C₄	—	2	2	no	—
372	C₄⋊C₄	—	4	2	no	—
373	C₄⋊C₄	—	8	4	no	—
374	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
375	C₂²⋊C₄	—	2	4	no	—
376	C₂²⋊C₄	—	2	4	no	—
377	C₂²⋊C₄	—	4	2	no	—
378	C₂²⋊C₄	—	8	4	no	—
379	C₂²⋊C₄	—	8	4	no	—
380	C₄²	—	2	4	no	—
381	C₄²	—	2	2	no	—
382	C₄²	—	2	2	no	—
383	C₄²	—	4	2	no	—
384	C₂³	—	2	2	no	—
385	C₂³	—	2	2	no	—
386	C₂³	—	2	4	no	—
387	C₂³	—	4	2	no	—
388	C₂³	—	4	2	no	—
389	C₂³	—	4	4	no	—
390	C₂³	—	4	4	no	—
391	C₂³	—	4	8	no	—
392	C₂³	—	8	8	no	—
393	C₂³	—	8	4	no	—
394	C₂³	—	8	4	no	—
395	C₂³	—	8	8	no	—
396	C₂³	C₄×D₄×C₆₇	1	2	yes	—
397	C₂³	#121×C₆₇	1	2	yes	—
398	C₂³	#124×C₆₇	1	2	yes	—
399	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
400	D₄	—	2	4	no	—
401	D₄	—	2	4	no	—
402	D₄	—	4	2	no	—
403	D₄	—	4	2	no	—
404	D₄	—	4	8	no	—
405	C₂×C₄	—	2	4	no	—
406	C₂×C₄	—	2	2	no	—
407	C₂×C₄	—	2	2	no	—
408	C₂×C₄	—	2	4	no	—
409	C₂×C₄	—	2	4	no	—
410	C₂×C₄	—	4	2	no	—
411	C₂×C₄	—	4	2	no	—
412	C₂×C₄	—	4	2	no	—
413	C₂×C₄	—	4	2	no	—
414	C₂×C₄	—	4	2	no	—
415	C₂×C₄	—	4	4	no	—
416	C₂×C₄	—	4	2	no	—
417	C₂×C₄	—	8	4	no	—
418	C₂×C₄	#117×C₆₇	1	4	yes	—
419	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
420	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
421	C₂×C₄	#121×C₆₇	1	2	yes	—
422	C₂×C₄	#123×C₆₇	1	4	yes	—
423	C₂×C₄	#124×C₆₇	1	2	yes	—
424	C₂×C₄	#126×C₆₇	1	4	yes	—
425	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
426	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
427	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
428	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
429	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
430	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
431	C₈	—	2	4	no	—
432	C₈	—	2	2	no	—
433	C₈	—	2	2	no	—
434	C₂²	—	2	2	no	—
435	C₂²	—	2	2	no	—
436	C₂²	—	2	4	no	—
437	C₂²	—	2	4	no	—
438	C₂²	—	4	4	no	—
439	C₂²	—	4	8	no	—
440	C₂²	—	8	16	no	—
441	C₂²	—	8	2	no	—
442	C₂²	—	8	2	no	—
443	C₂²	—	8	4	no	—
444	C₂²	#388×C₆₇	1	4	yes	—
445	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
446	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
447	C₂²	C₂×#124×C₆₇	1	1	yes	characteristic
448	C₂²	#576×C₆₇	1	2	yes	—
449	C₂²	#577×C₆₇	1	2	yes	—
450	C₄	—	4	2	no	—
451	C₄	—	4	2	no	—
452	C₄	—	4	4	no	—
453	C₄	—	8	2	no	—
454	C₄	—	8	2	no	—
455	C₄	#388×C₆₇	1	4	yes	—
456	C₄	C₂×#117×C₆₇	1	1	yes	characteristic
457	C₄	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
458	C₄	C₂×#123×C₆₇	1	1	yes	characteristic
459	C₄	C₂×#126×C₆₇	1	1	yes	characteristic
460	C₂	—	2	4	no	—
461	C₂	—	8	4	no	—
462	C₂	C₂×#388×C₆₇	1	1	yes	characteristic
463	C₂	C₆₇×#2946	1	2	yes	—
464	C₂	C₆₇×#3057	1	1	yes	characteristic
465	C₂	C₆₇×#3058	1	1	yes	characteristic
466	C₂	C₆₇×#3136	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂³×D₄×C₆₇	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×#117×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×#405×C₆₇	C₂	semidirect
4. C₆₇×#2317	C₂	semidirect
6. C₆₇×#1936	C₂	semidirect
7. C₆₇×#1934	C₂	semidirect
8. C₆₇×#1736	C₂	semidirect
9. C₆₇×#1735	C₂	semidirect
17. #418×C₆₇	C₄	semidirect
23. C₂×#97×C₆₇	C₂²	semidirect
24. C₂×C₄×C₈×C₆₇	C₂²	semidirect
28. #389×C₆₇	C₂²	semidirect
33. #340×C₆₇	C₂²	semidirect
34. #340×C₆₇	C₂²	semidirect
35. #340×C₆₇	C₂²	semidirect
81. #102×C₆₇	C₂×C₄	semidirect
226. #21091	C₆₇	direct (N × H)
236. C₂×#405	C₂×C₆₇	semidirect
237. #2317	C₂×C₆₇	semidirect
239. #1936	C₂×C₆₇	semidirect
240. #1934	C₂×C₆₇	semidirect
241. #1736	C₂×C₆₇	semidirect
242. #1735	C₂×C₆₇	semidirect
243. C₆₇	#21091	direct (N × H)
251. #418	C₄×C₆₇	semidirect
257. C₂×#97	C₂²×C₆₇	semidirect
258. C₂×C₄×C₈	C₂²×C₆₇	semidirect
262. #389	C₂²×C₆₇	semidirect
267. #340	C₂²×C₆₇	semidirect
268. #340	C₂²×C₆₇	semidirect
269. #340	C₂²×C₆₇	semidirect
315. #102	C₂×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×#102×C₆₇	2
C₂×#405×C₆₇	2
C₆₇×#2317	2
C₆₇×#2309	2
C₆₇×#1936	2
C₆₇×#1934	2
C₆₇×#1736	2
C₆₇×#1735	2
#21091	67

Subgroups — C67×#21091 #30700342

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21091 #30700342