Subgroups — C₄×#421×C₆₇ #30699655

Order 17152.

1618 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	C₂×C₄×#41×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#421×C₆₇	C₂	1	2	yes	maximal
3	C₂×C₄×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×#117×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1759	C₂	1	4	yes	maximal, complemented
6	C₄×#104×C₆₇	C₂	1	4	yes	maximal, complemented
7	C₆₇×#1733	C₂	1	2	yes	maximal, complemented
8	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
9	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
10	C₄×#41×C₆₇	—	2	2	no	—
11	C₄×#41×C₆₇	—	2	1	no	—
12	C₄×#41×C₆₇	—	2	1	no	—
13	C₄×#41×C₆₇	C₄	1	4	yes	complemented
14	C₄×#41×C₆₇	C₂²	1	2	yes	—
15	C₄×#41×C₆₇	C₂²	1	2	yes	—
16	C₂×C₄×Q₈×C₆₇	C₄	1	2	yes	complemented
17	C₂×C₄×D₄×C₆₇	—	2	2	no	—
18	C₂×C₄×D₄×C₆₇	C₄	1	2	yes	complemented
19	C₂×#117×C₆₇	—	2	1	no	—
20	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×#117×C₆₇	C₂²	1	2	yes	—
22	C₂²×C₄²×C₆₇	—	2	1	no	—
23	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
24	#421×C₆₇	C₄	1	16	yes	direct product factor
25	C₂×#104×C₆₇	C₂²	1	8	yes	—
26	C₄×M₁₆×C₆₇	—	2	1	no	—
27	C₄×M₁₆×C₆₇	—	2	1	no	—
28	C₄×M₁₆×C₆₇	C₂²	1	2	yes	complemented
29	C₄×M₁₆×C₆₇	C₂²	1	2	yes	complemented
30	C₂×#97×C₆₇	—	2	1	no	—
31	C₂×C₄×C₈×C₆₇	—	2	1	no	—
32	C₄×C₄⋊C₄×C₆₇	—	2	2	no	—
33	C₄×C₂²⋊C₄×C₆₇	—	2	2	no	—
34	#376×C₆₇	C₂²	1	2	yes	complemented
35	C₄³×C₆₇	C₂²	1	2	yes	complemented
36	#339×C₆₇	C₂²	1	8	yes	complemented
37	C₂×#41×C₆₇	—	2	1	no	—
38	C₂×#41×C₆₇	—	2	1	no	—
39	C₂×#41×C₆₇	—	2	4	no	—
40	C₂×#41×C₆₇	C₂×C₄	1	8	yes	—
41	C₂×#41×C₆₇	C₂×C₄	1	2	yes	—
42	C₂×#41×C₆₇	C₂³	1	2	yes	—
43	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
44	C₂²×D₄×C₆₇	—	2	1	no	—
45	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
46	C₂³×C₄×C₆₇	—	2	1	no	—
47	C₂³×C₄×C₆₇	—	2	2	no	—
48	C₂³×C₄×C₆₇	D₄	1	2	yes	—
49	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
50	C₂×M₁₆×C₆₇	—	2	2	no	—
51	C₂×M₁₆×C₆₇	—	2	4	no	—
52	C₂×M₁₆×C₆₇	C₂×C₄	1	8	yes	complemented
53	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
54	C₂²×C₈×C₆₇	—	2	2	no	—
55	C₄×Q₈×C₆₇	—	2	2	no	—
56	C₄×Q₈×C₆₇	—	2	2	no	—
57	C₄×Q₈×C₆₇	C₂×C₄	1	4	yes	—
58	C₄×Q₈×C₆₇	C₂×C₄	1	4	yes	—
59	C₄×D₄×C₆₇	—	2	4	no	—
60	C₄×D₄×C₆₇	—	2	4	no	—
61	C₄×D₄×C₆₇	—	2	4	no	—
62	C₄×D₄×C₆₇	—	2	4	no	—
63	C₄×D₄×C₆₇	—	2	2	no	—
64	C₄×D₄×C₆₇	—	2	2	no	—
65	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	—
66	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	—
67	#117×C₆₇	—	2	1	no	—
68	#117×C₆₇	—	2	1	no	—
69	#117×C₆₇	—	2	4	no	—
70	#117×C₆₇	—	2	4	no	—
71	#117×C₆₇	—	2	2	no	—
72	#117×C₆₇	—	2	2	no	—
73	#117×C₆₇	C₂×C₄	1	2	yes	—
74	#117×C₆₇	C₂×C₄	1	8	yes	complemented
75	#117×C₆₇	C₂³	1	2	yes	—
76	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
77	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
78	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
79	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
80	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
81	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
82	C₂×C₄²×C₆₇	—	2	2	no	—
83	C₂×C₄²×C₆₇	—	2	1	no	—
84	C₂×C₄²×C₆₇	—	2	4	no	—
85	C₂×C₄²×C₆₇	—	2	2	no	—
86	C₂×C₄²×C₆₇	—	2	2	no	—
87	C₂×C₄²×C₆₇	—	2	4	no	—
88	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
89	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
90	C₂×C₄²×C₆₇	D₄	1	2	yes	—
91	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
92	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
93	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
94	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
95	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
96	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
97	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
98	#104×C₆₇	C₂×C₄	1	32	yes	complemented
99	#97×C₆₇	—	2	2	no	—
100	#97×C₆₇	—	2	2	no	—
101	C₄×C₈×C₆₇	—	2	2	no	—
102	C₄×C₈×C₆₇	—	2	2	no	—
103	#95×C₆₇	—	2	4	no	—
104	C₂⁴×C₆₇	—	2	1	no	—
105	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
106	#41×C₆₇	—	2	4	no	—
107	#41×C₆₇	—	2	8	no	—
108	#41×C₆₇	—	2	8	no	—
109	#41×C₆₇	—	2	4	no	—
110	#41×C₆₇	C₄²	1	8	yes	complemented
111	#41×C₆₇	C₂²×C₄	1	8	yes	—
112	C₂×Q₈×C₆₇	—	2	1	no	—
113	C₂×Q₈×C₆₇	—	2	2	no	—
114	C₂×Q₈×C₆₇	C₄²	1	4	yes	complemented
115	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
116	C₂×D₄×C₆₇	—	2	2	no	—
117	C₂×D₄×C₆₇	—	2	8	no	—
118	C₂×D₄×C₆₇	—	2	2	no	—
119	C₂×D₄×C₆₇	—	2	1	no	—
120	C₂×D₄×C₆₇	—	2	2	no	—
121	C₂×D₄×C₆₇	C₄²	1	4	yes	complemented
122	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
123	C₂²×C₄×C₆₇	—	2	1	no	—
124	C₂²×C₄×C₆₇	—	2	2	no	—
125	C₂²×C₄×C₆₇	—	2	8	no	—
126	C₂²×C₄×C₆₇	—	2	2	no	—
127	C₂²×C₄×C₆₇	—	2	2	no	—
128	C₂²×C₄×C₆₇	—	2	4	no	—
129	C₂²×C₄×C₆₇	—	2	4	no	—
130	C₂²×C₄×C₆₇	—	2	4	no	—
131	C₂²×C₄×C₆₇	—	4	4	no	—
132	C₂²×C₄×C₆₇	—	4	8	no	—
133	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
134	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
135	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
136	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
137	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
138	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
139	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
140	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
141	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
142	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
143	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
144	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
145	C₂²×C₄×C₆₇	#41	1	4	yes	—
146	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
147	M₁₆×C₆₇	—	2	8	no	—
148	M₁₆×C₆₇	—	2	4	no	—
149	M₁₆×C₆₇	C₂²×C₄	1	8	yes	complemented
150	C₂×C₈×C₆₇	—	2	8	no	—
151	C₂×C₈×C₆₇	—	2	4	no	—
152	C₄⋊C₄×C₆₇	—	2	4	no	—
153	C₄⋊C₄×C₆₇	—	2	4	no	—
154	C₄⋊C₄×C₆₇	—	2	2	no	—
155	C₄⋊C₄×C₆₇	—	2	8	no	—
156	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
157	C₂²⋊C₄×C₆₇	—	2	2	no	—
158	C₂²⋊C₄×C₆₇	—	2	8	no	—
159	C₂²⋊C₄×C₆₇	—	2	2	no	—
160	C₂²⋊C₄×C₆₇	—	2	4	no	—
161	C₂²⋊C₄×C₆₇	—	2	4	no	—
162	C₄²×C₆₇	—	2	2	no	—
163	C₄²×C₆₇	—	2	4	no	—
164	C₄²×C₆₇	—	2	4	no	—
165	C₄²×C₆₇	—	2	2	no	—
166	C₄²×C₆₇	—	2	4	no	—
167	C₄²×C₆₇	—	4	8	no	—
168	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
169	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
170	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
171	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
172	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
173	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
174	C₄²×C₆₇	C₂²×C₄	1	8	yes	complemented
175	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
176	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
177	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
178	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
179	C₂³×C₆₇	—	2	2	no	—
180	C₂³×C₆₇	—	2	4	no	—
181	C₂³×C₆₇	—	4	2	no	—
182	C₂³×C₆₇	—	4	4	no	—
183	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
184	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
185	C₂³×C₆₇	C₄×D₄	1	4	yes	—
186	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
187	Q₈×C₆₇	—	2	2	no	—
188	Q₈×C₆₇	—	2	4	no	—
189	Q₈×C₆₇	C₂×C₄²	1	4	yes	—
190	D₄×C₆₇	—	2	8	no	—
191	D₄×C₆₇	—	2	8	no	—
192	D₄×C₆₇	—	2	2	no	—
193	D₄×C₆₇	—	2	4	no	—
194	D₄×C₆₇	C₂×C₄²	1	4	yes	—
195	C₂×C₄×C₆₇	—	2	2	no	—
196	C₂×C₄×C₆₇	—	2	4	no	—
197	C₂×C₄×C₆₇	—	2	8	no	—
198	C₂×C₄×C₆₇	—	2	8	no	—
199	C₂×C₄×C₆₇	—	2	4	no	—
200	C₂×C₄×C₆₇	—	2	4	no	—
201	C₂×C₄×C₆₇	—	2	4	no	—
202	C₂×C₄×C₆₇	—	2	4	no	—
203	C₂×C₄×C₆₇	—	2	8	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	—	4	8	no	—
208	C₂×C₄×C₆₇	—	4	8	no	—
209	C₂×C₄×C₆₇	—	4	4	no	—
210	C₂×C₄×C₆₇	—	4	8	no	—
211	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
212	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
213	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
214	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
215	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
216	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
217	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
218	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
219	C₂×C₄×C₆₇	#117	1	2	yes	—
220	C₂×C₄×C₆₇	#117	1	2	yes	—
221	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
222	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
223	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
224	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
225	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
226	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
227	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
228	C₈×C₆₇	—	2	8	no	—
229	C₂²×C₆₇	—	2	2	no	—
230	C₂²×C₆₇	—	2	2	no	—
231	C₂²×C₆₇	—	2	2	no	—
232	C₂²×C₆₇	—	2	4	no	—
233	C₂²×C₆₇	—	4	4	no	—
234	C₂²×C₆₇	—	4	8	no	—
235	C₂²×C₆₇	—	4	4	no	—
236	C₂²×C₆₇	C₄×C₂²⋊C₄	1	2	yes	—
237	C₂²×C₆₇	C₄×C₂²⋊C₄	1	2	yes	—
238	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
239	C₂²×C₆₇	C₂×C₄×D₄	1	2	yes	—
240	C₄×C₆₇	—	2	2	no	—
241	C₄×C₆₇	—	2	2	no	—
242	C₄×C₆₇	—	2	2	no	—
243	C₄×C₆₇	—	2	4	no	—
244	C₄×C₆₇	—	4	8	no	—
245	C₄×C₆₇	—	4	4	no	—
246	C₄×C₆₇	C₄×C₂²⋊C₄	1	2	yes	—
247	C₄×C₆₇	C₄×C₂²⋊C₄	1	2	yes	—
248	C₄×C₆₇	#421	1	4	yes	direct product factor
249	C₄×C₆₇	C₂²×C₄²	1	1	yes	characteristic
250	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
251	C₄×C₆₇	C₂×C₄×D₄	1	2	yes	—
252	C₄×#421	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
253	C₂×C₆₇	—	2	2	no	—
254	C₂×C₆₇	—	2	2	no	—
255	C₂×C₆₇	—	2	2	no	—
256	C₂×C₆₇	—	4	4	no	—
257	C₂×C₆₇	C₂×C₄×C₂²⋊C₄	1	1	yes	characteristic
258	C₂×C₆₇	C₂×#421	1	2	yes	—
259	C₂×C₄×#41	C₂×C₆₇	1	1	yes	characteristic
260	C₂×#421	C₂×C₆₇	1	2	yes	—
261	C₂×C₄×M₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
262	C₄×#117	C₂×C₆₇	1	1	yes	characteristic, complemented
263	#1759	C₂×C₆₇	1	4	yes	complemented
264	C₄×#104	C₂×C₆₇	1	4	yes	complemented
265	#1733	C₂×C₆₇	1	2	yes	complemented
266	C₆₇	C₄×#421	1	1	yes	characteristic, direct product factor, Sylow
267	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
268	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
269	C₄×#41	—	2	2	no	—
270	C₄×#41	—	2	1	no	—
271	C₄×#41	—	2	1	no	—
272	C₄×#41	C₄×C₆₇	1	4	yes	complemented
273	C₄×#41	C₂²×C₆₇	1	2	yes	—
274	C₄×#41	C₂²×C₆₇	1	2	yes	—
275	C₂×C₄×Q₈	C₄×C₆₇	1	2	yes	complemented
276	C₂×C₄×D₄	—	2	2	no	—
277	C₂×C₄×D₄	C₄×C₆₇	1	2	yes	complemented
278	C₂×#117	—	2	1	no	—
279	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
280	C₂×#117	C₂²×C₆₇	1	2	yes	—
281	C₂²×C₄²	—	2	1	no	—
282	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
283	#421	C₄×C₆₇	1	16	yes	direct product factor
284	C₂×#104	C₂²×C₆₇	1	8	yes	—
285	C₄×M₁₆	—	2	1	no	—
286	C₄×M₁₆	—	2	1	no	—
287	C₄×M₁₆	C₂²×C₆₇	1	2	yes	complemented
288	C₄×M₁₆	C₂²×C₆₇	1	2	yes	complemented
289	C₂×#97	—	2	1	no	—
290	C₂×C₄×C₈	—	2	1	no	—
291	C₄×C₄⋊C₄	—	2	2	no	—
292	C₄×C₂²⋊C₄	—	2	2	no	—
293	#376	C₂²×C₆₇	1	2	yes	complemented
294	C₄³	C₂²×C₆₇	1	2	yes	complemented
295	#339	C₂²×C₆₇	1	8	yes	complemented
296	C₂×#41	—	2	1	no	—
297	C₂×#41	—	2	1	no	—
298	C₂×#41	—	2	4	no	—
299	C₂×#41	C₂×C₄×C₆₇	1	8	yes	—
300	C₂×#41	C₂×C₄×C₆₇	1	2	yes	—
301	C₂×#41	C₂³×C₆₇	1	2	yes	—
302	C₂²×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
303	C₂²×D₄	—	2	1	no	—
304	C₂²×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
305	C₂³×C₄	—	2	1	no	—
306	C₂³×C₄	—	2	2	no	—
307	C₂³×C₄	D₄×C₆₇	1	2	yes	—
308	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
309	C₂×M₁₆	—	2	2	no	—
310	C₂×M₁₆	—	2	4	no	—
311	C₂×M₁₆	C₂×C₄×C₆₇	1	8	yes	complemented
312	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
313	C₂²×C₈	—	2	2	no	—
314	C₄×Q₈	—	2	2	no	—
315	C₄×Q₈	—	2	2	no	—
316	C₄×Q₈	C₂×C₄×C₆₇	1	4	yes	—
317	C₄×Q₈	C₂×C₄×C₆₇	1	4	yes	—
318	C₄×D₄	—	2	4	no	—
319	C₄×D₄	—	2	4	no	—
320	C₄×D₄	—	2	4	no	—
321	C₄×D₄	—	2	4	no	—
322	C₄×D₄	—	2	2	no	—
323	C₄×D₄	—	2	2	no	—
324	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	—
325	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	—
326	#117	—	2	1	no	—
327	#117	—	2	1	no	—
328	#117	—	2	4	no	—
329	#117	—	2	4	no	—
330	#117	—	2	2	no	—
331	#117	—	2	2	no	—
332	#117	C₂×C₄×C₆₇	1	2	yes	—
333	#117	C₂×C₄×C₆₇	1	8	yes	complemented
334	#117	C₂³×C₆₇	1	2	yes	—
335	C₂×C₄⋊C₄	—	2	2	no	—
336	C₂×C₄⋊C₄	—	2	2	no	—
337	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
338	C₂×C₂²⋊C₄	—	2	1	no	—
339	C₂×C₂²⋊C₄	—	2	2	no	—
340	C₂×C₂²⋊C₄	—	2	2	no	—
341	C₂×C₄²	—	2	2	no	—
342	C₂×C₄²	—	2	1	no	—
343	C₂×C₄²	—	2	4	no	—
344	C₂×C₄²	—	2	2	no	—
345	C₂×C₄²	—	2	2	no	—
346	C₂×C₄²	—	2	4	no	—
347	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
348	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
349	C₂×C₄²	D₄×C₆₇	1	2	yes	—
350	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
351	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
352	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
353	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
354	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
355	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
356	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
357	#104	C₂×C₄×C₆₇	1	32	yes	complemented
358	#97	—	2	2	no	—
359	#97	—	2	2	no	—
360	C₄×C₈	—	2	2	no	—
361	C₄×C₈	—	2	2	no	—
362	#95	—	2	4	no	—
363	C₂⁴	—	2	1	no	—
364	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
365	#41	—	2	4	no	—
366	#41	—	2	8	no	—
367	#41	—	2	8	no	—
368	#41	—	2	4	no	—
369	#41	C₄²×C₆₇	1	8	yes	complemented
370	#41	C₂²×C₄×C₆₇	1	8	yes	—
371	C₂×Q₈	—	2	1	no	—
372	C₂×Q₈	—	2	2	no	—
373	C₂×Q₈	C₄²×C₆₇	1	4	yes	complemented
374	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
375	C₂×D₄	—	2	2	no	—
376	C₂×D₄	—	2	8	no	—
377	C₂×D₄	—	2	2	no	—
378	C₂×D₄	—	2	1	no	—
379	C₂×D₄	—	2	2	no	—
380	C₂×D₄	C₄²×C₆₇	1	4	yes	complemented
381	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
382	C₂²×C₄	—	2	1	no	—
383	C₂²×C₄	—	2	2	no	—
384	C₂²×C₄	—	2	8	no	—
385	C₂²×C₄	—	2	2	no	—
386	C₂²×C₄	—	2	2	no	—
387	C₂²×C₄	—	2	4	no	—
388	C₂²×C₄	—	2	4	no	—
389	C₂²×C₄	—	2	4	no	—
390	C₂²×C₄	—	4	4	no	—
391	C₂²×C₄	—	4	8	no	—
392	C₂²×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
393	C₂²×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
394	C₂²×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
395	C₂²×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
396	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
397	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
398	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
399	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
400	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
401	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
402	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
403	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
404	C₂²×C₄	#41×C₆₇	1	4	yes	—
405	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
406	M₁₆	—	2	8	no	—
407	M₁₆	—	2	4	no	—
408	M₁₆	C₂²×C₄×C₆₇	1	8	yes	complemented
409	C₂×C₈	—	2	8	no	—
410	C₂×C₈	—	2	4	no	—
411	C₄⋊C₄	—	2	4	no	—
412	C₄⋊C₄	—	2	4	no	—
413	C₄⋊C₄	—	2	2	no	—
414	C₄⋊C₄	—	2	8	no	—
415	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
416	C₂²⋊C₄	—	2	2	no	—
417	C₂²⋊C₄	—	2	8	no	—
418	C₂²⋊C₄	—	2	2	no	—
419	C₂²⋊C₄	—	2	4	no	—
420	C₂²⋊C₄	—	2	4	no	—
421	C₄²	—	2	2	no	—
422	C₄²	—	2	4	no	—
423	C₄²	—	2	4	no	—
424	C₄²	—	2	2	no	—
425	C₄²	—	2	4	no	—
426	C₄²	—	4	8	no	—
427	C₄²	C₂²⋊C₄×C₆₇	1	2	yes	—
428	C₄²	C₂²⋊C₄×C₆₇	1	2	yes	—
429	C₄²	C₂²⋊C₄×C₆₇	1	2	yes	—
430	C₄²	C₂²⋊C₄×C₆₇	1	2	yes	—
431	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
432	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
433	C₄²	C₂²×C₄×C₆₇	1	8	yes	complemented
434	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
435	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
436	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
437	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
438	C₂³	—	2	2	no	—
439	C₂³	—	2	4	no	—
440	C₂³	—	4	2	no	—
441	C₂³	—	4	4	no	—
442	C₂³	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
443	C₂³	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
444	C₂³	C₄×D₄×C₆₇	1	4	yes	—
445	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
446	Q₈	—	2	2	no	—
447	Q₈	—	2	4	no	—
448	Q₈	C₂×C₄²×C₆₇	1	4	yes	—
449	D₄	—	2	8	no	—
450	D₄	—	2	8	no	—
451	D₄	—	2	2	no	—
452	D₄	—	2	4	no	—
453	D₄	C₂×C₄²×C₆₇	1	4	yes	—
454	C₂×C₄	—	2	2	no	—
455	C₂×C₄	—	2	4	no	—
456	C₂×C₄	—	2	8	no	—
457	C₂×C₄	—	2	8	no	—
458	C₂×C₄	—	2	4	no	—
459	C₂×C₄	—	2	4	no	—
460	C₂×C₄	—	2	4	no	—
461	C₂×C₄	—	2	4	no	—
462	C₂×C₄	—	2	8	no	—
463	C₂×C₄	—	4	2	no	—
464	C₂×C₄	—	4	2	no	—
465	C₂×C₄	—	4	2	no	—
466	C₂×C₄	—	4	8	no	—
467	C₂×C₄	—	4	8	no	—
468	C₂×C₄	—	4	4	no	—
469	C₂×C₄	—	4	8	no	—
470	C₂×C₄	C₂×C₄²×C₆₇	1	2	yes	—
471	C₂×C₄	C₂×C₄²×C₆₇	1	2	yes	—
472	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
473	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
474	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
475	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
476	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
477	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
478	C₂×C₄	#117×C₆₇	1	2	yes	—
479	C₂×C₄	#117×C₆₇	1	2	yes	—
480	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
481	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
482	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
483	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
484	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
485	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
486	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
487	C₈	—	2	8	no	—
488	C₂²	—	2	2	no	—
489	C₂²	—	2	2	no	—
490	C₂²	—	2	2	no	—
491	C₂²	—	2	4	no	—
492	C₂²	—	4	4	no	—
493	C₂²	—	4	8	no	—
494	C₂²	—	4	4	no	—
495	C₂²	C₄×C₂²⋊C₄×C₆₇	1	2	yes	—
496	C₂²	C₄×C₂²⋊C₄×C₆₇	1	2	yes	—
497	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
498	C₂²	C₂×C₄×D₄×C₆₇	1	2	yes	—
499	C₄	—	2	4	no	—
500	C₄	—	2	2	no	—
501	C₄	—	2	2	no	—
502	C₄	—	2	2	no	—
503	C₄	—	4	8	no	—
504	C₄	—	4	4	no	—
505	C₄	C₄×C₂²⋊C₄×C₆₇	1	2	yes	—
506	C₄	C₄×C₂²⋊C₄×C₆₇	1	2	yes	—
507	C₄	#421×C₆₇	1	4	yes	direct product factor
508	C₄	C₂²×C₄²×C₆₇	1	1	yes	characteristic
509	C₄	C₂×#117×C₆₇	1	1	yes	characteristic
510	C₄	C₂×C₄×D₄×C₆₇	1	2	yes	—
511	C₂	—	2	2	no	—
512	C₂	—	2	2	no	—
513	C₂	—	2	2	no	—
514	C₂	—	4	4	no	—
515	C₂	C₂×C₄×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
516	C₂	C₂×#421×C₆₇	1	2	yes	—

	subgroup	G/H
center	C₄²×C₆₇	C₂×D₄
commutator	C₄	C₂²×C₄²×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂²×C₂²⋊C₄
prime radical	C₂²×#41×C₆₇	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₄×M₁₆×C₆₇	C₂	semidirect
4. C₄×#117×C₆₇	C₂	semidirect
5. C₆₇×#1759	C₂	semidirect
6. C₄×#104×C₆₇	C₂	semidirect
7. C₆₇×#1733	C₂	semidirect
13. C₄×#41×C₆₇	C₄	semidirect
16. C₂×C₄×Q₈×C₆₇	C₄	semidirect
18. C₂×C₄×D₄×C₆₇	C₄	semidirect
24. #421×C₆₇	C₄	direct (N × H)
28. C₄×M₁₆×C₆₇	C₂²	semidirect
29. C₄×M₁₆×C₆₇	C₂²	semidirect
34. #376×C₆₇	C₂²	semidirect
35. C₄³×C₆₇	C₂²	semidirect
36. #339×C₆₇	C₂²	semidirect
52. C₂×M₁₆×C₆₇	C₂×C₄	semidirect
74. #117×C₆₇	C₂×C₄	semidirect
98. #104×C₆₇	C₂×C₄	semidirect
110. #41×C₆₇	C₄²	semidirect
114. C₂×Q₈×C₆₇	C₄²	semidirect
121. C₂×D₄×C₆₇	C₄²	semidirect
149. M₁₆×C₆₇	C₂²×C₄	semidirect
174. C₄²×C₆₇	C₂²×C₄	semidirect
248. C₄×C₆₇	#421	direct (N × H)
252. C₄×#421	C₆₇	direct (N × H)
261. C₂×C₄×M₁₆	C₂×C₆₇	semidirect
262. C₄×#117	C₂×C₆₇	semidirect
263. #1759	C₂×C₆₇	semidirect
264. C₄×#104	C₂×C₆₇	semidirect
265. #1733	C₂×C₆₇	semidirect
266. C₆₇	C₄×#421	direct (N × H)
272. C₄×#41	C₄×C₆₇	semidirect
275. C₂×C₄×Q₈	C₄×C₆₇	semidirect
277. C₂×C₄×D₄	C₄×C₆₇	semidirect
283. #421	C₄×C₆₇	direct (N × H)
287. C₄×M₁₆	C₂²×C₆₇	semidirect
288. C₄×M₁₆	C₂²×C₆₇	semidirect
293. #376	C₂²×C₆₇	semidirect
294. C₄³	C₂²×C₆₇	semidirect
295. #339	C₂²×C₆₇	semidirect
311. C₂×M₁₆	C₂×C₄×C₆₇	semidirect
333. #117	C₂×C₄×C₆₇	semidirect
357. #104	C₂×C₄×C₆₇	semidirect
369. #41	C₄²×C₆₇	semidirect
373. C₂×Q₈	C₄²×C₆₇	semidirect
380. C₂×D₄	C₄²×C₆₇	semidirect
408. M₁₆	C₂²×C₄×C₆₇	semidirect
433. C₄²	C₂²×C₄×C₆₇	semidirect
507. C₄	#421×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×C₄×#41×C₆₇	2
C₂×#421×C₆₇	2
C₂×C₄×M₁₆×C₆₇	2
C₄×#117×C₆₇	2
C₆₇×#1759	2
C₄×#104×C₆₇	2
C₆₇×#1733	2
C₄×#421	67

Subgroups — C4×#421×C67 #30699655

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#421×C₆₇ #30699655