Subgroups — C₄×#466×C₆₇ #30700772

Order 17152.

1222 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#466×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×D₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₄×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×#121×C₆₇	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1970	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1942	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1934	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1893	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1808	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#107×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#102×C₆₇	C₂	1	2	yes	maximal, complemented
12	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×#121×C₆₇	C₂²	1	2	yes	—
16	C₂×C₄×D₄×C₆₇	—	2	2	no	—
17	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
18	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
19	#466×C₆₇	C₄	1	8	yes	direct product factor
20	C₄×D₈×C₆₇	—	2	4	no	—
21	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×#102×C₆₇	C₂²	1	2	yes	—
25	C₂×#102×C₆₇	C₂²	1	2	yes	—
26	C₂×#102×C₆₇	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₆₇	—	2	4	no	—
28	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
32	#390×C₆₇	C₂²	1	2	yes	—
33	#389×C₆₇	C₂²	1	2	yes	—
34	#388×C₆₇	—	2	2	no	—
35	#386×C₆₇	—	2	2	no	—
36	#380×C₆₇	C₂²	1	2	yes	—
37	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
38	C₄×C₂²⋊C₄×C₆₇	—	2	2	no	—
39	#340×C₆₇	C₂²	1	2	yes	—
40	#340×C₆₇	C₂²	1	2	yes	—
41	#340×C₆₇	C₂²	1	2	yes	—
42	C₂²×D₄×C₆₇	—	2	2	no	—
43	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
44	C₂³×C₄×C₆₇	—	2	2	no	—
45	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
46	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
47	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
48	C₂×D₈×C₆₇	—	2	2	no	—
49	C₂×D₈×C₆₇	—	2	2	no	—
50	C₂×D₈×C₆₇	C₂×C₄	1	4	yes	complemented
51	C₂²×C₈×C₆₇	—	2	2	no	—
52	C₂²×C₈×C₆₇	—	2	2	no	—
53	C₂²×C₈×C₆₇	—	2	2	no	—
54	C₂²×C₈×C₆₇	—	2	2	no	—
55	C₂²×C₈×C₆₇	—	2	2	no	—
56	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	complemented
57	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
58	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
59	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
60	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
61	#121×C₆₇	—	2	4	no	—
62	#121×C₆₇	C₂×C₄	1	8	yes	complemented
63	C₄×D₄×C₆₇	—	2	4	no	—
64	C₄×D₄×C₆₇	—	2	4	no	—
65	C₄×D₄×C₆₇	—	4	8	no	—
66	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
67	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
68	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
69	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
70	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
71	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
72	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
73	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
74	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
75	C₂×C₄²×C₆₇	—	2	2	no	—
76	C₂×C₄²×C₆₇	—	2	2	no	—
77	C₂×C₄²×C₆₇	—	2	2	no	—
78	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
79	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
80	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
81	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
82	#107×C₆₇	—	2	2	no	—
83	#107×C₆₇	—	2	2	no	—
84	#107×C₆₇	C₂×C₄	1	4	yes	complemented
85	#102×C₆₇	—	2	4	no	—
86	#102×C₆₇	—	2	4	no	—
87	#102×C₆₇	C₂×C₄	1	8	yes	complemented
88	C₄×C₈×C₆₇	—	2	2	no	—
89	C₄×C₈×C₆₇	—	2	2	no	—
90	C₄×C₈×C₆₇	—	2	2	no	—
91	C₄×C₈×C₆₇	D₄	1	4	yes	complemented
92	#95×C₆₇	—	2	2	no	—
93	#95×C₆₇	—	2	2	no	—
94	#95×C₆₇	—	2	2	no	—
95	C₂⁴×C₆₇	—	2	2	no	—
96	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
97	C₂×D₄×C₆₇	—	2	8	no	—
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	—	4	4	no	—
104	C₂×D₄×C₆₇	—	4	4	no	—
105	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
106	C₂²×C₄×C₆₇	—	2	2	no	—
107	C₂²×C₄×C₆₇	—	2	2	no	—
108	C₂²×C₄×C₆₇	—	2	2	no	—
109	C₂²×C₄×C₆₇	—	2	2	no	—
110	C₂²×C₄×C₆₇	—	2	2	no	—
111	C₂²×C₄×C₆₇	—	2	1	no	—
112	C₂²×C₄×C₆₇	—	2	1	no	—
113	C₂²×C₄×C₆₇	—	2	1	no	—
114	C₂²×C₄×C₆₇	—	2	1	no	—
115	C₂²×C₄×C₆₇	—	2	2	no	—
116	C₂²×C₄×C₆₇	—	2	2	no	—
117	C₂²×C₄×C₆₇	—	2	2	no	—
118	C₂²×C₄×C₆₇	—	4	4	no	—
119	C₂²×C₄×C₆₇	—	4	4	no	—
120	C₂²×C₄×C₆₇	—	4	4	no	—
121	C₂²×C₄×C₆₇	D₈	1	4	yes	complemented
122	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
123	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
124	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
125	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
126	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
127	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
128	C₂²×C₄×C₆₇	#41	1	2	yes	—
129	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
130	D₈×C₆₇	—	2	8	no	—
131	C₂×C₈×C₆₇	—	2	8	no	—
132	C₂×C₈×C₆₇	—	2	1	no	—
133	C₂×C₈×C₆₇	—	2	1	no	—
134	C₂×C₈×C₆₇	—	2	1	no	—
135	C₂×C₈×C₆₇	—	2	1	no	—
136	C₂×C₈×C₆₇	—	2	1	no	—
137	C₂×C₈×C₆₇	—	2	1	no	—
138	C₂×C₈×C₆₇	—	2	8	no	—
139	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
140	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
141	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
142	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
143	C₂×C₈×C₆₇	#41	1	2	yes	—
144	C₂×C₈×C₆₇	#41	1	2	yes	—
145	C₄⋊C₄×C₆₇	—	2	2	no	—
146	C₄⋊C₄×C₆₇	—	2	2	no	—
147	C₄⋊C₄×C₆₇	—	2	2	no	—
148	C₄⋊C₄×C₆₇	—	2	2	no	—
149	C₄⋊C₄×C₆₇	—	2	2	no	—
150	C₄⋊C₄×C₆₇	—	4	4	no	—
151	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
152	C₂²⋊C₄×C₆₇	—	2	8	no	—
153	C₂²⋊C₄×C₆₇	—	4	4	no	—
154	C₂²⋊C₄×C₆₇	—	4	4	no	—
155	C₂²⋊C₄×C₆₇	—	4	4	no	—
156	C₂²⋊C₄×C₆₇	—	4	4	no	—
157	C₄²×C₆₇	—	2	2	no	—
158	C₄²×C₆₇	—	2	2	no	—
159	C₄²×C₆₇	—	2	1	no	—
160	C₄²×C₆₇	—	2	1	no	—
161	C₄²×C₆₇	—	4	4	no	—
162	C₄²×C₆₇	C₂×D₄	1	2	yes	—
163	C₄²×C₆₇	#41	1	2	yes	—
164	C₂³×C₆₇	—	2	1	no	—
165	C₂³×C₆₇	—	2	1	no	—
166	C₂³×C₆₇	—	2	1	no	—
167	C₂³×C₆₇	—	2	1	no	—
168	C₂³×C₆₇	—	2	4	no	—
169	C₂³×C₆₇	—	4	2	no	—
170	C₂³×C₆₇	—	4	2	no	—
171	C₂³×C₆₇	—	4	2	no	—
172	C₂³×C₆₇	—	4	2	no	—
173	C₂³×C₆₇	—	4	2	no	—
174	C₂³×C₆₇	—	4	2	no	—
175	C₂³×C₆₇	C₄×D₄	1	2	yes	—
176	C₂³×C₆₇	C₂×D₈	1	2	yes	—
177	C₂³×C₆₇	#135	1	2	yes	—
178	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
179	D₄×C₆₇	—	2	4	no	—
180	D₄×C₆₇	—	2	4	no	—
181	D₄×C₆₇	—	4	16	no	—
182	D₄×C₆₇	—	4	2	no	—
183	D₄×C₆₇	—	4	2	no	—
184	C₂×C₄×C₆₇	—	2	8	no	—
185	C₂×C₄×C₆₇	—	2	1	no	—
186	C₂×C₄×C₆₇	—	2	1	no	—
187	C₂×C₄×C₆₇	—	2	1	no	—
188	C₂×C₄×C₆₇	—	2	1	no	—
189	C₂×C₄×C₆₇	—	2	1	no	—
190	C₂×C₄×C₆₇	—	2	1	no	—
191	C₂×C₄×C₆₇	—	2	1	no	—
192	C₂×C₄×C₆₇	—	2	1	no	—
193	C₂×C₄×C₆₇	—	2	2	no	—
194	C₂×C₄×C₆₇	—	2	2	no	—
195	C₂×C₄×C₆₇	—	2	2	no	—
196	C₂×C₄×C₆₇	—	2	2	no	—
197	C₂×C₄×C₆₇	—	2	8	no	—
198	C₂×C₄×C₆₇	—	2	4	no	—
199	C₂×C₄×C₆₇	—	4	2	no	—
200	C₂×C₄×C₆₇	—	4	2	no	—
201	C₂×C₄×C₆₇	—	4	2	no	—
202	C₂×C₄×C₆₇	—	4	2	no	—
203	C₂×C₄×C₆₇	—	4	2	no	—
204	C₂×C₄×C₆₇	—	4	2	no	—
205	C₂×C₄×C₆₇	—	4	2	no	—
206	C₂×C₄×C₆₇	—	4	2	no	—
207	C₂×C₄×C₆₇	—	8	8	no	—
208	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
209	C₂×C₄×C₆₇	#121	1	2	yes	—
210	C₂×C₄×C₆₇	#121	1	2	yes	—
211	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
212	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
213	C₂×C₄×C₆₇	#135	1	2	yes	—
214	C₂×C₄×C₆₇	#135	1	2	yes	—
215	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
216	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
217	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
218	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
219	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
220	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
221	C₈×C₆₇	—	2	2	no	—
222	C₈×C₆₇	—	2	2	no	—
223	C₈×C₆₇	—	2	2	no	—
224	C₈×C₆₇	C₄×D₄	1	4	yes	complemented
225	C₂²×C₆₇	—	2	2	no	—
226	C₂²×C₆₇	—	2	2	no	—
227	C₂²×C₆₇	—	2	2	no	—
228	C₂²×C₆₇	—	2	2	no	—
229	C₂²×C₆₇	—	2	2	no	—
230	C₂²×C₆₇	—	2	2	no	—
231	C₂²×C₆₇	—	4	4	no	—
232	C₂²×C₆₇	—	4	4	no	—
233	C₂²×C₆₇	—	4	4	no	—
234	C₂²×C₆₇	—	8	2	no	—
235	C₂²×C₆₇	—	8	2	no	—
236	C₂²×C₆₇	—	8	2	no	—
237	C₂²×C₆₇	—	8	2	no	—
238	C₂²×C₆₇	C₄×D₈	1	4	yes	complemented
239	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
240	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
241	C₂²×C₆₇	#532	1	1	yes	characteristic
242	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
243	C₂²×C₆₇	C₂×#135	1	1	yes	characteristic
244	C₂²×C₆₇	C₂×#135	1	1	yes	characteristic
245	C₂²×C₆₇	C₂×#135	1	1	yes	characteristic
246	C₄×C₆₇	—	2	2	no	—
247	C₄×C₆₇	—	2	2	no	—
248	C₄×C₆₇	—	2	2	no	—
249	C₄×C₆₇	—	2	2	no	—
250	C₄×C₆₇	—	4	4	no	—
251	C₄×C₆₇	—	8	2	no	—
252	C₄×C₆₇	—	8	2	no	—
253	C₄×C₆₇	#466	1	4	yes	direct product factor
254	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
255	C₄×C₆₇	C₄×#41	1	1	yes	characteristic
256	C₄×C₆₇	#525	1	1	yes	characteristic
257	C₄×C₆₇	#529	1	1	yes	characteristic
258	C₄×#466	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
259	C₂×C₆₇	—	2	4	no	—
260	C₂×C₆₇	—	8	4	no	—
261	C₂×C₆₇	C₄×#121	1	1	yes	characteristic
262	C₂×C₆₇	C₂×C₄×D₈	1	1	yes	characteristic
263	C₂×C₆₇	C₄×#135	1	1	yes	characteristic
264	C₂×C₆₇	C₂×#466	1	1	yes	characteristic
265	C₂×C₆₇	#3050	1	1	yes	characteristic
266	C₂×C₆₇	#3168	1	1	yes	characteristic
267	C₂×C₆₇	#3169	1	1	yes	characteristic
268	C₂×#466	C₂×C₆₇	1	1	yes	characteristic
269	C₂×C₄×D₈	C₂×C₆₇	1	1	yes	characteristic, complemented
270	C₂²×C₄×C₈	C₂×C₆₇	1	1	yes	characteristic, complemented
271	C₄×#121	C₂×C₆₇	1	2	yes	complemented
272	#1970	C₂×C₆₇	1	1	yes	characteristic, complemented
273	#1942	C₂×C₆₇	1	1	yes	characteristic, complemented
274	#1934	C₂×C₆₇	1	2	yes	complemented
275	#1893	C₂×C₆₇	1	2	yes	complemented
276	#1808	C₂×C₆₇	1	1	yes	characteristic, complemented
277	C₄×#107	C₂×C₆₇	1	1	yes	characteristic, complemented
278	C₄×#102	C₂×C₆₇	1	2	yes	complemented
279	C₆₇	C₄×#466	1	1	yes	characteristic, direct product factor, Sylow
280	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
281	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
282	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
283	C₂×#121	C₂²×C₆₇	1	2	yes	—
284	C₂×C₄×D₄	—	2	2	no	—
285	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
286	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
287	#466	C₄×C₆₇	1	8	yes	direct product factor
288	C₄×D₈	—	2	4	no	—
289	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
290	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
291	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
292	C₂×#102	C₂²×C₆₇	1	2	yes	—
293	C₂×#102	C₂²×C₆₇	1	2	yes	—
294	C₂×#102	C₂²×C₆₇	1	2	yes	—
295	C₂×C₄×C₈	—	2	4	no	—
296	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
297	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
298	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
299	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
300	#390	C₂²×C₆₇	1	2	yes	—
301	#389	C₂²×C₆₇	1	2	yes	—
302	#388	—	2	2	no	—
303	#386	—	2	2	no	—
304	#380	C₂²×C₆₇	1	2	yes	—
305	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
306	C₄×C₂²⋊C₄	—	2	2	no	—
307	#340	C₂²×C₆₇	1	2	yes	—
308	#340	C₂²×C₆₇	1	2	yes	—
309	#340	C₂²×C₆₇	1	2	yes	—
310	C₂²×D₄	—	2	2	no	—
311	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
312	C₂³×C₄	—	2	2	no	—
313	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
314	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
315	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
316	C₂×D₈	—	2	2	no	—
317	C₂×D₈	—	2	2	no	—
318	C₂×D₈	C₂×C₄×C₆₇	1	4	yes	complemented
319	C₂²×C₈	—	2	2	no	—
320	C₂²×C₈	—	2	2	no	—
321	C₂²×C₈	—	2	2	no	—
322	C₂²×C₈	—	2	2	no	—
323	C₂²×C₈	—	2	2	no	—
324	C₂²×C₈	C₂×C₄×C₆₇	1	4	yes	complemented
325	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
326	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
327	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
328	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
329	#121	—	2	4	no	—
330	#121	C₂×C₄×C₆₇	1	8	yes	complemented
331	C₄×D₄	—	2	4	no	—
332	C₄×D₄	—	2	4	no	—
333	C₄×D₄	—	4	8	no	—
334	C₂×C₄⋊C₄	—	2	2	no	—
335	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
336	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
337	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
338	C₂×C₂²⋊C₄	—	2	2	no	—
339	C₂×C₂²⋊C₄	—	2	2	no	—
340	C₂×C₂²⋊C₄	—	2	2	no	—
341	C₂×C₂²⋊C₄	—	2	2	no	—
342	C₂×C₂²⋊C₄	—	2	2	no	—
343	C₂×C₄²	—	2	2	no	—
344	C₂×C₄²	—	2	2	no	—
345	C₂×C₄²	—	2	2	no	—
346	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
347	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
348	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
349	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
350	#107	—	2	2	no	—
351	#107	—	2	2	no	—
352	#107	C₂×C₄×C₆₇	1	4	yes	complemented
353	#102	—	2	4	no	—
354	#102	—	2	4	no	—
355	#102	C₂×C₄×C₆₇	1	8	yes	complemented
356	C₄×C₈	—	2	2	no	—
357	C₄×C₈	—	2	2	no	—
358	C₄×C₈	—	2	2	no	—
359	C₄×C₈	D₄×C₆₇	1	4	yes	complemented
360	#95	—	2	2	no	—
361	#95	—	2	2	no	—
362	#95	—	2	2	no	—
363	C₂⁴	—	2	2	no	—
364	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
365	C₂×D₄	—	2	8	no	—
366	C₂×D₄	—	2	2	no	—
367	C₂×D₄	—	2	2	no	—
368	C₂×D₄	—	2	2	no	—
369	C₂×D₄	—	2	2	no	—
370	C₂×D₄	—	2	2	no	—
371	C₂×D₄	—	4	4	no	—
372	C₂×D₄	—	4	4	no	—
373	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
374	C₂²×C₄	—	2	2	no	—
375	C₂²×C₄	—	2	2	no	—
376	C₂²×C₄	—	2	2	no	—
377	C₂²×C₄	—	2	2	no	—
378	C₂²×C₄	—	2	2	no	—
379	C₂²×C₄	—	2	1	no	—
380	C₂²×C₄	—	2	1	no	—
381	C₂²×C₄	—	2	1	no	—
382	C₂²×C₄	—	2	1	no	—
383	C₂²×C₄	—	2	2	no	—
384	C₂²×C₄	—	2	2	no	—
385	C₂²×C₄	—	2	2	no	—
386	C₂²×C₄	—	4	4	no	—
387	C₂²×C₄	—	4	4	no	—
388	C₂²×C₄	—	4	4	no	—
389	C₂²×C₄	D₈×C₆₇	1	4	yes	complemented
390	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
391	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
392	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
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	#41×C₆₇	1	2	yes	—
397	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
398	D₈	—	2	8	no	—
399	C₂×C₈	—	2	8	no	—
400	C₂×C₈	—	2	1	no	—
401	C₂×C₈	—	2	1	no	—
402	C₂×C₈	—	2	1	no	—
403	C₂×C₈	—	2	1	no	—
404	C₂×C₈	—	2	1	no	—
405	C₂×C₈	—	2	1	no	—
406	C₂×C₈	—	2	8	no	—
407	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
408	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
409	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
410	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
411	C₂×C₈	#41×C₆₇	1	2	yes	—
412	C₂×C₈	#41×C₆₇	1	2	yes	—
413	C₄⋊C₄	—	2	2	no	—
414	C₄⋊C₄	—	2	2	no	—
415	C₄⋊C₄	—	2	2	no	—
416	C₄⋊C₄	—	2	2	no	—
417	C₄⋊C₄	—	2	2	no	—
418	C₄⋊C₄	—	4	4	no	—
419	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
420	C₂²⋊C₄	—	2	8	no	—
421	C₂²⋊C₄	—	4	4	no	—
422	C₂²⋊C₄	—	4	4	no	—
423	C₂²⋊C₄	—	4	4	no	—
424	C₂²⋊C₄	—	4	4	no	—
425	C₄²	—	2	2	no	—
426	C₄²	—	2	2	no	—
427	C₄²	—	2	1	no	—
428	C₄²	—	2	1	no	—
429	C₄²	—	4	4	no	—
430	C₄²	C₂×D₄×C₆₇	1	2	yes	—
431	C₄²	#41×C₆₇	1	2	yes	—
432	C₂³	—	2	1	no	—
433	C₂³	—	2	1	no	—
434	C₂³	—	2	1	no	—
435	C₂³	—	2	1	no	—
436	C₂³	—	2	4	no	—
437	C₂³	—	4	2	no	—
438	C₂³	—	4	2	no	—
439	C₂³	—	4	2	no	—
440	C₂³	—	4	2	no	—
441	C₂³	—	4	2	no	—
442	C₂³	—	4	2	no	—
443	C₂³	C₄×D₄×C₆₇	1	2	yes	—
444	C₂³	C₂×D₈×C₆₇	1	2	yes	—
445	C₂³	#135×C₆₇	1	2	yes	—
446	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
447	D₄	—	2	4	no	—
448	D₄	—	2	4	no	—
449	D₄	—	4	16	no	—
450	D₄	—	4	2	no	—
451	D₄	—	4	2	no	—
452	C₂×C₄	—	2	8	no	—
453	C₂×C₄	—	2	1	no	—
454	C₂×C₄	—	2	1	no	—
455	C₂×C₄	—	2	1	no	—
456	C₂×C₄	—	2	1	no	—
457	C₂×C₄	—	2	1	no	—
458	C₂×C₄	—	2	1	no	—
459	C₂×C₄	—	2	1	no	—
460	C₂×C₄	—	2	1	no	—
461	C₂×C₄	—	2	8	no	—
462	C₂×C₄	—	2	2	no	—
463	C₂×C₄	—	2	2	no	—
464	C₂×C₄	—	2	2	no	—
465	C₂×C₄	—	2	2	no	—
466	C₂×C₄	—	2	4	no	—
467	C₂×C₄	—	4	2	no	—
468	C₂×C₄	—	4	2	no	—
469	C₂×C₄	—	4	2	no	—
470	C₂×C₄	—	4	2	no	—
471	C₂×C₄	—	4	2	no	—
472	C₂×C₄	—	4	2	no	—
473	C₂×C₄	—	4	2	no	—
474	C₂×C₄	—	4	2	no	—
475	C₂×C₄	—	8	8	no	—
476	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
477	C₂×C₄	#121×C₆₇	1	2	yes	—
478	C₂×C₄	#121×C₆₇	1	2	yes	—
479	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
480	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
481	C₂×C₄	#135×C₆₇	1	2	yes	—
482	C₂×C₄	#135×C₆₇	1	2	yes	—
483	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
484	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
485	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
486	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
487	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
488	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
489	C₈	—	2	2	no	—
490	C₈	—	2	2	no	—
491	C₈	—	2	2	no	—
492	C₈	C₄×D₄×C₆₇	1	4	yes	complemented
493	C₂²	—	2	2	no	—
494	C₂²	—	2	2	no	—
495	C₂²	—	2	2	no	—
496	C₂²	—	2	2	no	—
497	C₂²	—	2	2	no	—
498	C₂²	—	2	2	no	—
499	C₂²	—	4	4	no	—
500	C₂²	—	4	4	no	—
501	C₂²	—	4	4	no	—
502	C₂²	—	8	2	no	—
503	C₂²	—	8	2	no	—
504	C₂²	—	8	2	no	—
505	C₂²	—	8	2	no	—
506	C₂²	C₄×D₈×C₆₇	1	4	yes	complemented
507	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
508	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
509	C₂²	#532×C₆₇	1	1	yes	characteristic
510	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
511	C₂²	C₂×#135×C₆₇	1	1	yes	characteristic
512	C₂²	C₂×#135×C₆₇	1	1	yes	characteristic
513	C₂²	C₂×#135×C₆₇	1	1	yes	characteristic
514	C₄	—	2	2	no	—
515	C₄	—	2	2	no	—
516	C₄	—	2	2	no	—
517	C₄	—	2	2	no	—
518	C₄	—	4	4	no	—
519	C₄	—	8	2	no	—
520	C₄	—	8	2	no	—
521	C₄	#466×C₆₇	1	4	yes	direct product factor
522	C₄	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
523	C₄	C₄×#41×C₆₇	1	1	yes	characteristic
524	C₄	#525×C₆₇	1	1	yes	characteristic
525	C₄	#529×C₆₇	1	1	yes	characteristic
526	C₂	—	2	4	no	—
527	C₂	—	8	4	no	—
528	C₂	C₄×#121×C₆₇	1	1	yes	characteristic
529	C₂	C₂×C₄×D₈×C₆₇	1	1	yes	characteristic
530	C₂	C₄×#135×C₆₇	1	1	yes	characteristic
531	C₂	C₂×#466×C₆₇	1	1	yes	characteristic
532	C₂	C₆₇×#3050	1	1	yes	characteristic
533	C₂	C₆₇×#3168	1	1	yes	characteristic
534	C₂	C₆₇×#3169	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂³×C₆₇	C₂²×D₄
prime radical	C₂×#466×C₆₇	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×D₈×C₆₇	C₂	semidirect
3. C₂²×C₄×C₈×C₆₇	C₂	semidirect
4. C₄×#121×C₆₇	C₂	semidirect
5. C₆₇×#1970	C₂	semidirect
6. C₆₇×#1942	C₂	semidirect
7. C₆₇×#1934	C₂	semidirect
8. C₆₇×#1893	C₂	semidirect
9. C₆₇×#1808	C₂	semidirect
10. C₄×#107×C₆₇	C₂	semidirect
11. C₄×#102×C₆₇	C₂	semidirect
19. #466×C₆₇	C₄	direct (N × H)
50. C₂×D₈×C₆₇	C₂×C₄	semidirect
56. C₂²×C₈×C₆₇	C₂×C₄	semidirect
62. #121×C₆₇	C₂×C₄	semidirect
84. #107×C₆₇	C₂×C₄	semidirect
87. #102×C₆₇	C₂×C₄	semidirect
91. C₄×C₈×C₆₇	D₄	semidirect
121. C₂²×C₄×C₆₇	D₈	semidirect
224. C₈×C₆₇	C₄×D₄	semidirect
238. C₂²×C₆₇	C₄×D₈	semidirect
253. C₄×C₆₇	#466	direct (N × H)
258. C₄×#466	C₆₇	direct (N × H)
269. C₂×C₄×D₈	C₂×C₆₇	semidirect
270. C₂²×C₄×C₈	C₂×C₆₇	semidirect
271. C₄×#121	C₂×C₆₇	semidirect
272. #1970	C₂×C₆₇	semidirect
273. #1942	C₂×C₆₇	semidirect
274. #1934	C₂×C₆₇	semidirect
275. #1893	C₂×C₆₇	semidirect
276. #1808	C₂×C₆₇	semidirect
277. C₄×#107	C₂×C₆₇	semidirect
278. C₄×#102	C₂×C₆₇	semidirect
279. C₆₇	C₄×#466	direct (N × H)
287. #466	C₄×C₆₇	direct (N × H)
318. C₂×D₈	C₂×C₄×C₆₇	semidirect
324. C₂²×C₈	C₂×C₄×C₆₇	semidirect
330. #121	C₂×C₄×C₆₇	semidirect
352. #107	C₂×C₄×C₆₇	semidirect
355. #102	C₂×C₄×C₆₇	semidirect
359. C₄×C₈	D₄×C₆₇	semidirect
389. C₂²×C₄	D₈×C₆₇	semidirect
492. C₈	C₄×D₄×C₆₇	semidirect
506. C₂²	C₄×D₈×C₆₇	semidirect
521. C₄	#466×C₆₇	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×#466×C₆₇	2
C₂×C₄×D₈×C₆₇	2
C₂²×C₄×C₈×C₆₇	2
C₄×#121×C₆₇	2
C₆₇×#1970	2
C₆₇×#1942	2
C₆₇×#1934	2
C₆₇×#1893	2
C₆₇×#1808	2
C₄×#107×C₆₇	2
C₄×#102×C₆₇	2
C₄×#466	67

Subgroups — C4×#466×C67 #30700772

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#466×C₆₇ #30700772