Subgroups — C₆₇×#22511 #30701762

Order 17152.

1036 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#533×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#446×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₆₇×#1985	C₂	1	2	yes	maximal, complemented
4	C₆₇×#1985	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1984	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1798	C₂	1	2	yes	maximal
7	C₆₇×#1797	C₂	1	1	yes	maximal, characteristic
8	C₆₇×#1750	C₂	1	2	yes	maximal
9	C₆₇×#1749	C₂	1	1	yes	maximal, characteristic
10	#533×C₆₇	—	2	8	no	—
11	C₂×C₄⋊Q₈×C₆₇	C₄	1	1	yes	characteristic
12	C₂×C₄⋊Q₈×C₆₇	C₄	1	1	yes	characteristic
13	C₂×#125×C₆₇	C₄	1	2	yes	—
14	C₂×C₂²⋊Q₈×C₆₇	C₄	1	1	yes	characteristic
15	C₂×C₂²⋊Q₈×C₆₇	C₄	1	1	yes	characteristic
16	C₂×C₂²⋊Q₈×C₆₇	C₄	1	2	yes	—
17	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
19	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×#117×C₆₇	C₂²	1	2	yes	—
21	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
22	#446×C₆₇	—	2	8	no	—
23	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
24	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
25	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
26	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
28	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×#97×C₆₇	C₂²	1	2	yes	—
30	C₂×#97×C₆₇	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
32	#336×C₆₇	C₂²	1	2	yes	—
33	#336×C₆₇	C₂²	1	1	yes	characteristic
34	#336×C₆₇	C₂²	1	2	yes	—
35	#336×C₆₇	C₂²	1	2	yes	—
36	#336×C₆₇	C₂²	1	2	yes	—
37	#336×C₆₇	C₂²	1	2	yes	—
38	#336×C₆₇	C₂²	1	1	yes	characteristic
39	C₂²×Q₈×C₆₇	C₂×C₄	1	2	yes	—
40	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
41	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
42	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
43	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
44	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
46	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
47	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
48	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
49	C₄⋊Q₈×C₆₇	—	2	4	no	—
50	C₄⋊Q₈×C₆₇	—	2	4	no	—
51	#125×C₆₇	—	2	8	no	—
52	C₂²⋊Q₈×C₆₇	—	2	4	no	—
53	C₂²⋊Q₈×C₆₇	—	2	4	no	—
54	C₂²⋊Q₈×C₆₇	—	2	8	no	—
55	C₄×Q₈×C₆₇	—	2	4	no	—
56	C₄×Q₈×C₆₇	—	2	4	no	—
57	C₄×Q₈×C₆₇	—	2	4	no	—
58	C₄×Q₈×C₆₇	D₄	1	8	yes	—
59	#117×C₆₇	—	2	4	no	—
60	#117×C₆₇	—	2	4	no	—
61	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
62	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
63	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
64	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
65	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
66	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
68	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
69	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
70	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
71	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
72	C₂×C₄²×C₆₇	—	2	4	no	—
73	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
74	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
75	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
76	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
77	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
78	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
79	C₄⋊C₈×C₆₇	—	2	4	no	—
80	C₄⋊C₈×C₆₇	—	2	4	no	—
81	C₄⋊C₈×C₆₇	—	2	4	no	—
82	C₂²⋊C₈×C₆₇	—	2	2	no	—
83	C₂²⋊C₈×C₆₇	—	2	4	no	—
84	C₂²⋊C₈×C₆₇	Q₈	1	4	yes	complemented
85	#97×C₆₇	—	2	4	no	—
86	#97×C₆₇	—	2	4	no	—
87	C₄×C₈×C₆₇	—	2	4	no	—
88	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
89	C₂×Q₈×C₆₇	—	2	4	no	—
90	C₂×Q₈×C₆₇	—	2	2	no	—
91	C₂×Q₈×C₆₇	—	2	4	no	—
92	C₂×Q₈×C₆₇	—	4	1	no	—
93	C₂×Q₈×C₆₇	—	4	1	no	—
94	C₂×Q₈×C₆₇	—	4	2	no	—
95	C₂×Q₈×C₆₇	—	4	1	no	—
96	C₂×Q₈×C₆₇	—	4	1	no	—
97	C₂×Q₈×C₆₇	C₂²⋊C₄	1	4	yes	—
98	C₂²×C₄×C₆₇	—	2	2	no	—
99	C₂²×C₄×C₆₇	—	2	2	no	—
100	C₂²×C₄×C₆₇	—	2	2	no	—
101	C₂²×C₄×C₆₇	—	2	2	no	—
102	C₂²×C₄×C₆₇	—	2	2	no	—
103	C₂²×C₄×C₆₇	—	4	1	no	—
104	C₂²×C₄×C₆₇	—	4	1	no	—
105	C₂²×C₄×C₆₇	—	4	1	no	—
106	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
107	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
108	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
109	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
110	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
111	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
112	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
113	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
114	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
115	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
116	C₂²×C₄×C₆₇	#41	1	2	yes	—
117	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
118	C₂×C₈×C₆₇	—	2	2	no	—
119	C₂×C₈×C₆₇	—	2	2	no	—
120	C₂×C₈×C₆₇	—	2	2	no	—
121	C₂×C₈×C₆₇	—	2	2	no	—
122	C₂×C₈×C₆₇	—	2	2	no	—
123	C₂×C₈×C₆₇	—	2	2	no	—
124	C₂×C₈×C₆₇	—	2	2	no	—
125	C₂×C₈×C₆₇	—	2	2	no	—
126	C₂×C₈×C₆₇	—	2	2	no	—
127	C₂×C₈×C₆₇	—	2	2	no	—
128	C₂×C₈×C₆₇	—	2	2	no	—
129	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
130	C₄⋊C₄×C₆₇	—	2	4	no	—
131	C₄⋊C₄×C₆₇	—	2	4	no	—
132	C₄⋊C₄×C₆₇	—	2	4	no	—
133	C₄⋊C₄×C₆₇	—	2	4	no	—
134	C₄⋊C₄×C₆₇	—	2	4	no	—
135	C₄⋊C₄×C₆₇	—	2	4	no	—
136	C₄⋊C₄×C₆₇	—	2	2	no	—
137	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	8	yes	—
138	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
139	C₂²⋊C₄×C₆₇	—	2	4	no	—
140	C₂²⋊C₄×C₆₇	—	2	4	no	—
141	C₄²×C₆₇	—	2	2	no	—
142	C₄²×C₆₇	—	2	2	no	—
143	C₄²×C₆₇	—	2	2	no	—
144	C₄²×C₆₇	—	2	2	no	—
145	C₄²×C₆₇	—	2	2	no	—
146	C₄²×C₆₇	C₂×D₄	1	4	yes	—
147	C₄²×C₆₇	C₂×D₄	1	4	yes	—
148	C₄²×C₆₇	C₂×D₄	1	4	yes	—
149	C₂³×C₆₇	—	2	1	no	—
150	C₂³×C₆₇	—	2	1	no	—
151	C₂³×C₆₇	—	2	1	no	—
152	C₂³×C₆₇	—	2	1	no	—
153	C₂³×C₆₇	—	2	1	no	—
154	C₂³×C₆₇	—	2	1	no	—
155	C₂³×C₆₇	C₄×Q₈	1	2	yes	—
156	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
157	Q₈×C₆₇	—	4	4	no	—
158	Q₈×C₆₇	—	4	8	no	—
159	Q₈×C₆₇	—	4	4	no	—
160	C₂×C₄×C₆₇	—	2	1	no	—
161	C₂×C₄×C₆₇	—	2	1	no	—
162	C₂×C₄×C₆₇	—	2	1	no	—
163	C₂×C₄×C₆₇	—	2	1	no	—
164	C₂×C₄×C₆₇	—	2	2	no	—
165	C₂×C₄×C₆₇	—	2	1	no	—
166	C₂×C₄×C₆₇	—	2	1	no	—
167	C₂×C₄×C₆₇	—	2	1	no	—
168	C₂×C₄×C₆₇	—	2	1	no	—
169	C₂×C₄×C₆₇	—	2	2	no	—
170	C₂×C₄×C₆₇	—	2	2	no	—
171	C₂×C₄×C₆₇	—	2	2	no	—
172	C₂×C₄×C₆₇	—	2	4	no	—
173	C₂×C₄×C₆₇	—	2	4	no	—
174	C₂×C₄×C₆₇	—	2	1	no	—
175	C₂×C₄×C₆₇	—	2	2	no	—
176	C₂×C₄×C₆₇	—	2	2	no	—
177	C₂×C₄×C₆₇	—	2	2	no	—
178	C₂×C₄×C₆₇	—	2	1	no	—
179	C₂×C₄×C₆₇	—	2	1	no	—
180	C₂×C₄×C₆₇	—	2	2	no	—
181	C₂×C₄×C₆₇	—	2	2	no	—
182	C₂×C₄×C₆₇	—	2	2	no	—
183	C₂×C₄×C₆₇	—	4	4	no	—
184	C₂×C₄×C₆₇	—	4	4	no	—
185	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
186	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
187	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
188	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
189	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
190	C₂×C₄×C₆₇	#131	1	2	yes	—
191	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
192	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
193	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
194	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
195	C₈×C₆₇	—	2	4	no	—
196	C₈×C₆₇	—	4	2	no	—
197	C₈×C₆₇	—	4	2	no	—
198	C₈×C₆₇	—	4	2	no	—
199	C₂²×C₆₇	—	2	2	no	—
200	C₂²×C₆₇	—	2	2	no	—
201	C₂²×C₆₇	—	2	2	no	—
202	C₂²×C₆₇	—	4	1	no	—
203	C₂²×C₆₇	—	4	1	no	—
204	C₂²×C₆₇	—	4	1	no	—
205	C₂²×C₆₇	—	4	1	no	—
206	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
207	C₂²×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
208	C₂²×C₆₇	#519	1	1	yes	characteristic
209	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
210	C₂²×C₆₇	#568	1	1	yes	characteristic
211	C₂²×C₆₇	#568	1	1	yes	characteristic
212	C₂²×C₆₇	#568	1	1	yes	characteristic
213	C₄×C₆₇	—	2	2	no	—
214	C₄×C₆₇	—	2	2	no	—
215	C₄×C₆₇	—	2	4	no	—
216	C₄×C₆₇	—	2	2	no	—
217	C₄×C₆₇	—	2	2	no	—
218	C₄×C₆₇	—	2	4	no	—
219	C₄×C₆₇	—	4	1	no	—
220	C₄×C₆₇	—	4	2	no	—
221	C₄×C₆₇	—	4	2	no	—
222	C₄×C₆₇	#548	1	2	yes	—
223	C₄×C₆₇	D₄×Q₈	1	2	yes	—
224	#22511	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
225	C₂×C₆₇	—	4	2	no	—
226	C₂×C₆₇	Q₈×C₂²⋊C₄	1	1	yes	characteristic
227	C₂×C₆₇	#2879	1	1	yes	characteristic
228	C₂×C₆₇	#2880	1	1	yes	characteristic
229	C₂×C₆₇	#2961	1	1	yes	characteristic
230	C₂×C₆₇	#2962	1	1	yes	characteristic
231	C₂×C₆₇	#2985	1	1	yes	characteristic
232	C₂×C₆₇	#2987	1	1	yes	characteristic
233	C₂×#533	C₂×C₆₇	1	1	yes	characteristic
234	C₂×#446	C₂×C₆₇	1	2	yes	complemented
235	#1985	C₂×C₆₇	1	2	yes	complemented
236	#1985	C₂×C₆₇	1	2	yes	complemented
237	#1984	C₂×C₆₇	1	2	yes	complemented
238	#1798	C₂×C₆₇	1	2	yes	—
239	#1797	C₂×C₆₇	1	1	yes	characteristic
240	#1750	C₂×C₆₇	1	2	yes	—
241	#1749	C₂×C₆₇	1	1	yes	characteristic
242	C₆₇	#22511	1	1	yes	characteristic, direct product factor, Sylow
243	#533	—	2	8	no	—
244	C₂×C₄⋊Q₈	C₄×C₆₇	1	1	yes	characteristic
245	C₂×C₄⋊Q₈	C₄×C₆₇	1	1	yes	characteristic
246	C₂×#125	C₄×C₆₇	1	2	yes	—
247	C₂×C₂²⋊Q₈	C₄×C₆₇	1	1	yes	characteristic
248	C₂×C₂²⋊Q₈	C₄×C₆₇	1	1	yes	characteristic
249	C₂×C₂²⋊Q₈	C₄×C₆₇	1	2	yes	—
250	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
251	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
252	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
253	C₂×#117	C₂²×C₆₇	1	2	yes	—
254	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
255	#446	—	2	8	no	—
256	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
257	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
258	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
259	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
260	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
261	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
262	C₂×#97	C₂²×C₆₇	1	2	yes	—
263	C₂×#97	C₂²×C₆₇	1	2	yes	—
264	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
265	#336	C₂²×C₆₇	1	1	yes	characteristic
266	#336	C₂²×C₆₇	1	1	yes	characteristic
267	#336	C₂²×C₆₇	1	2	yes	—
268	#336	C₂²×C₆₇	1	2	yes	—
269	#336	C₂²×C₆₇	1	2	yes	—
270	#336	C₂²×C₆₇	1	2	yes	—
271	#336	C₂²×C₆₇	1	2	yes	—
272	C₂²×Q₈	C₂×C₄×C₆₇	1	2	yes	—
273	C₂²×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
274	C₂²×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
275	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
276	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
277	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
278	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
279	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
280	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
281	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
282	C₄⋊Q₈	—	2	4	no	—
283	C₄⋊Q₈	—	2	4	no	—
284	#125	—	2	8	no	—
285	C₂²⋊Q₈	—	2	4	no	—
286	C₂²⋊Q₈	—	2	4	no	—
287	C₂²⋊Q₈	—	2	8	no	—
288	C₄×Q₈	—	2	4	no	—
289	C₄×Q₈	—	2	4	no	—
290	C₄×Q₈	—	2	4	no	—
291	C₄×Q₈	D₄×C₆₇	1	8	yes	—
292	#117	—	2	4	no	—
293	#117	—	2	4	no	—
294	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
295	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
296	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
297	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
298	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
299	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
300	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
301	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
302	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
303	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
304	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
305	C₂×C₄²	—	2	4	no	—
306	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
307	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
308	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
309	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
310	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
311	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
312	C₄⋊C₈	—	2	4	no	—
313	C₄⋊C₈	—	2	4	no	—
314	C₄⋊C₈	—	2	4	no	—
315	C₂²⋊C₈	—	2	4	no	—
316	C₂²⋊C₈	—	2	2	no	—
317	C₂²⋊C₈	Q₈×C₆₇	1	4	yes	complemented
318	#97	—	2	4	no	—
319	#97	—	2	4	no	—
320	C₄×C₈	—	2	4	no	—
321	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
322	C₂×Q₈	—	2	4	no	—
323	C₂×Q₈	—	2	2	no	—
324	C₂×Q₈	—	2	4	no	—
325	C₂×Q₈	—	4	1	no	—
326	C₂×Q₈	—	4	1	no	—
327	C₂×Q₈	—	4	2	no	—
328	C₂×Q₈	—	4	1	no	—
329	C₂×Q₈	—	4	1	no	—
330	C₂×Q₈	C₂²⋊C₄×C₆₇	1	4	yes	—
331	C₂²×C₄	—	2	2	no	—
332	C₂²×C₄	—	2	2	no	—
333	C₂²×C₄	—	2	2	no	—
334	C₂²×C₄	—	2	2	no	—
335	C₂²×C₄	—	2	2	no	—
336	C₂²×C₄	—	4	1	no	—
337	C₂²×C₄	—	4	1	no	—
338	C₂²×C₄	—	4	1	no	—
339	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
340	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
341	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
342	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
343	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
344	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
345	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
346	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
347	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
348	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
349	C₂²×C₄	#41×C₆₇	1	2	yes	—
350	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
351	C₂×C₈	—	2	2	no	—
352	C₂×C₈	—	2	2	no	—
353	C₂×C₈	—	2	2	no	—
354	C₂×C₈	—	2	2	no	—
355	C₂×C₈	—	2	2	no	—
356	C₂×C₈	—	2	2	no	—
357	C₂×C₈	—	2	2	no	—
358	C₂×C₈	—	2	2	no	—
359	C₂×C₈	—	2	2	no	—
360	C₂×C₈	—	2	2	no	—
361	C₂×C₈	—	2	2	no	—
362	C₂×C₈	C₂×Q₈×C₆₇	1	4	yes	—
363	C₄⋊C₄	—	2	4	no	—
364	C₄⋊C₄	—	2	4	no	—
365	C₄⋊C₄	—	2	4	no	—
366	C₄⋊C₄	—	2	4	no	—
367	C₄⋊C₄	—	2	4	no	—
368	C₄⋊C₄	—	2	4	no	—
369	C₄⋊C₄	—	2	2	no	—
370	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	8	yes	—
371	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
372	C₂²⋊C₄	—	2	4	no	—
373	C₂²⋊C₄	—	2	4	no	—
374	C₄²	—	2	2	no	—
375	C₄²	—	2	2	no	—
376	C₄²	—	2	2	no	—
377	C₄²	—	2	2	no	—
378	C₄²	—	2	2	no	—
379	C₄²	C₂×D₄×C₆₇	1	4	yes	—
380	C₄²	C₂×D₄×C₆₇	1	4	yes	—
381	C₄²	C₂×D₄×C₆₇	1	4	yes	—
382	C₂³	—	2	1	no	—
383	C₂³	—	2	1	no	—
384	C₂³	—	2	1	no	—
385	C₂³	—	2	1	no	—
386	C₂³	—	2	1	no	—
387	C₂³	—	2	1	no	—
388	C₂³	C₄×Q₈×C₆₇	1	2	yes	—
389	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
390	Q₈	—	4	4	no	—
391	Q₈	—	4	8	no	—
392	Q₈	—	4	4	no	—
393	C₂×C₄	—	2	1	no	—
394	C₂×C₄	—	2	1	no	—
395	C₂×C₄	—	2	1	no	—
396	C₂×C₄	—	2	1	no	—
397	C₂×C₄	—	2	2	no	—
398	C₂×C₄	—	2	1	no	—
399	C₂×C₄	—	2	1	no	—
400	C₂×C₄	—	2	1	no	—
401	C₂×C₄	—	2	1	no	—
402	C₂×C₄	—	2	2	no	—
403	C₂×C₄	—	2	2	no	—
404	C₂×C₄	—	2	2	no	—
405	C₂×C₄	—	2	4	no	—
406	C₂×C₄	—	2	4	no	—
407	C₂×C₄	—	2	1	no	—
408	C₂×C₄	—	2	2	no	—
409	C₂×C₄	—	2	2	no	—
410	C₂×C₄	—	2	2	no	—
411	C₂×C₄	—	2	1	no	—
412	C₂×C₄	—	2	1	no	—
413	C₂×C₄	—	2	2	no	—
414	C₂×C₄	—	2	2	no	—
415	C₂×C₄	—	2	2	no	—
416	C₂×C₄	—	4	4	no	—
417	C₂×C₄	—	4	4	no	—
418	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
419	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
420	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
421	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
422	C₂×C₄	C₄×Q₈×C₆₇	1	2	yes	—
423	C₂×C₄	#131×C₆₇	1	2	yes	—
424	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
425	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
426	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
427	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
428	C₈	—	2	4	no	—
429	C₈	—	4	2	no	—
430	C₈	—	4	2	no	—
431	C₈	—	4	2	no	—
432	C₂²	—	2	2	no	—
433	C₂²	—	2	2	no	—
434	C₂²	—	2	2	no	—
435	C₂²	—	4	1	no	—
436	C₂²	—	4	1	no	—
437	C₂²	—	4	1	no	—
438	C₂²	—	4	1	no	—
439	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
440	C₂²	C₂×C₄×Q₈×C₆₇	1	1	yes	characteristic
441	C₂²	#519×C₆₇	1	1	yes	characteristic
442	C₂²	C₂×#131×C₆₇	1	1	yes	characteristic
443	C₂²	#568×C₆₇	1	1	yes	characteristic
444	C₂²	#568×C₆₇	1	1	yes	characteristic
445	C₂²	#568×C₆₇	1	1	yes	characteristic
446	C₄	—	2	2	no	—
447	C₄	—	2	2	no	—
448	C₄	—	2	4	no	—
449	C₄	—	2	2	no	—
450	C₄	—	2	2	no	—
451	C₄	—	2	4	no	—
452	C₄	—	4	1	no	—
453	C₄	—	4	2	no	—
454	C₄	—	4	2	no	—
455	C₄	#548×C₆₇	1	2	yes	—
456	C₄	D₄×Q₈×C₆₇	1	2	yes	—
457	C₂	—	4	2	no	—
458	C₂	Q₈×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
459	C₂	C₆₇×#2879	1	1	yes	characteristic
460	C₂	C₆₇×#2880	1	1	yes	characteristic
461	C₂	C₆₇×#2961	1	1	yes	characteristic
462	C₂	C₆₇×#2962	1	1	yes	characteristic
463	C₂	C₆₇×#2985	1	1	yes	characteristic
464	C₂	C₆₇×#2987	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂×#446×C₆₇	C₂	semidirect
3. C₆₇×#1985	C₂	semidirect
4. C₆₇×#1985	C₂	semidirect
5. C₆₇×#1984	C₂	semidirect
84. C₂²⋊C₈×C₆₇	Q₈	semidirect
224. #22511	C₆₇	direct (N × H)
234. C₂×#446	C₂×C₆₇	semidirect
235. #1985	C₂×C₆₇	semidirect
236. #1985	C₂×C₆₇	semidirect
237. #1984	C₂×C₆₇	semidirect
242. C₆₇	#22511	direct (N × H)
317. C₂²⋊C₈	Q₈×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#533×C₆₇	2
C₂×#446×C₆₇	2
C₆₇×#1985	2
C₆₇×#1984	2
C₆₇×#1798	2
C₆₇×#1797	2
C₆₇×#1750	2
C₆₇×#1749	2
#22511	67

Subgroups — C67×#22511 #30701762

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22511 #30701762