Subgroups — C₆₇×#22696 #30701947

Order 17152.

1108 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₄×C₆₇	C₂	1	2	yes	maximal
2	C₄²×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#1939	C₂	1	2	yes	maximal, complemented
4	C₆₇×#1850	C₂	1	4	yes	maximal, complemented
5	C₆₇×#1831	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1797	C₂	1	2	yes	maximal, complemented
7	C₄×C₄⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#336×C₆₇	C₂	1	2	yes	maximal, complemented
9	C₂³×C₈×C₆₇	C₂²	1	4	yes	—
10	C₂×C₄×D₄×C₆₇	C₄	1	1	yes	characteristic
11	C₂×C₄×D₄×C₆₇	C₄	1	1	yes	characteristic
12	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
13	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
14	C₈×D₄×C₆₇	—	2	8	no	—
15	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
17	C₂×C₄⋊C₈×C₆₇	C₂²	1	4	yes	—
18	C₂×C₂²⋊C₈×C₆₇	C₂²	1	4	yes	—
19	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
20	C₄×C₄⋊C₄×C₆₇	C₄	1	1	yes	characteristic
21	C₄×C₄⋊C₄×C₆₇	C₄	1	1	yes	characteristic
22	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
23	C₄×C₂²⋊C₄×C₆₇	C₄	1	2	yes	—
24	C₄×C₂²⋊C₄×C₆₇	C₄	1	2	yes	—
25	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
26	C₄³×C₆₇	C₂²	1	1	yes	characteristic
27	#336×C₆₇	C₂²	1	1	yes	characteristic
28	#336×C₆₇	C₂²	1	1	yes	characteristic
29	#336×C₆₇	C₂²	1	2	yes	—
30	#336×C₆₇	C₂²	1	2	yes	—
31	#336×C₆₇	C₂²	1	4	yes	—
32	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
33	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
34	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
35	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
36	C₂²×C₈×C₆₇	—	2	8	no	—
37	C₂²×C₈×C₆₇	—	2	8	no	—
38	C₂²×C₈×C₆₇	—	4	4	no	—
39	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
40	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
41	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
42	C₄×D₄×C₆₇	—	2	4	no	—
43	C₄×D₄×C₆₇	C₈	1	8	yes	complemented
44	C₄×D₄×C₆₇	D₄	1	4	yes	—
45	C₄×D₄×C₆₇	Q₈	1	4	yes	—
46	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
49	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
50	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
51	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₆₇	—	4	4	no	—
53	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
56	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
57	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
60	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
61	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
62	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
63	C₄⋊C₈×C₆₇	—	2	4	no	—
64	C₄⋊C₈×C₆₇	—	2	4	no	—
65	C₄⋊C₈×C₆₇	—	4	4	no	—
66	C₂²⋊C₈×C₆₇	—	2	8	no	—
67	C₄×C₈×C₆₇	—	2	4	no	—
68	#95×C₆₇	C₂×C₄	1	2	yes	—
69	#95×C₆₇	C₂×C₄	1	2	yes	—
70	#95×C₆₇	C₂×C₄	1	2	yes	—
71	C₂⁴×C₆₇	C₂²×C₄	1	2	yes	—
72	C₂×D₄×C₆₇	C₄⋊C₄	1	4	yes	—
73	C₂×D₄×C₆₇	C₂×C₈	1	4	yes	—
74	C₂×D₄×C₆₇	M₁₆	1	4	yes	—
75	C₂²×C₄×C₆₇	—	2	4	no	—
76	C₂²×C₄×C₆₇	—	2	4	no	—
77	C₂²×C₄×C₆₇	—	2	4	no	—
78	C₂²×C₄×C₆₇	—	2	4	no	—
79	C₂²×C₄×C₆₇	—	4	2	no	—
80	C₂²×C₄×C₆₇	—	4	2	no	—
81	C₂²×C₄×C₆₇	—	4	2	no	—
82	C₂²×C₄×C₆₇	C₂×C₈	1	8	yes	complemented
83	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
85	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
87	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
90	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
93	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
94	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
95	C₂×C₈×C₆₇	—	2	2	no	—
96	C₂×C₈×C₆₇	—	2	2	no	—
97	C₂×C₈×C₆₇	—	2	4	no	—
98	C₂×C₈×C₆₇	—	2	2	no	—
99	C₂×C₈×C₆₇	—	2	2	no	—
100	C₂×C₈×C₆₇	—	2	4	no	—
101	C₂×C₈×C₆₇	—	2	2	no	—
102	C₂×C₈×C₆₇	—	2	4	no	—
103	C₂×C₈×C₆₇	—	4	16	no	—
104	C₂×C₈×C₆₇	#41	1	4	yes	—
105	C₄⋊C₄×C₆₇	—	2	2	no	—
106	C₄⋊C₄×C₆₇	C₄⋊C₄	1	4	yes	—
107	C₄⋊C₄×C₆₇	C₂×C₈	1	4	yes	complemented
108	C₂²⋊C₄×C₆₇	—	2	4	no	—
109	C₂²⋊C₄×C₆₇	C₄⋊C₄	1	8	yes	—
110	C₂²⋊C₄×C₆₇	C₂×C₈	1	8	yes	complemented
111	C₄²×C₆₇	—	2	2	no	—
112	C₄²×C₆₇	—	2	2	no	—
113	C₄²×C₆₇	—	2	2	no	—
114	C₄²×C₆₇	—	2	4	no	—
115	C₄²×C₆₇	—	2	2	no	—
116	C₄²×C₆₇	—	4	2	no	—
117	C₄²×C₆₇	—	4	2	no	—
118	C₄²×C₆₇	—	4	1	no	—
119	C₄²×C₆₇	C₂×C₈	1	4	yes	complemented
120	C₄²×C₆₇	C₂×D₄	1	2	yes	—
121	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
122	C₂³×C₆₇	—	2	2	no	—
123	C₂³×C₆₇	—	2	2	no	—
124	C₂³×C₆₇	—	2	2	no	—
125	C₂³×C₆₇	—	2	2	no	—
126	C₂³×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
127	C₂³×C₆₇	C₂²×C₈	1	4	yes	—
128	C₂³×C₆₇	C₂×M₁₆	1	4	yes	—
129	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
130	D₄×C₆₇	C₄⋊C₈	1	16	yes	complemented
131	C₂×C₄×C₆₇	—	2	1	no	—
132	C₂×C₄×C₆₇	—	2	2	no	—
133	C₂×C₄×C₆₇	—	2	1	no	—
134	C₂×C₄×C₆₇	—	2	1	no	—
135	C₂×C₄×C₆₇	—	2	2	no	—
136	C₂×C₄×C₆₇	—	2	2	no	—
137	C₂×C₄×C₆₇	—	2	1	no	—
138	C₂×C₄×C₆₇	—	2	1	no	—
139	C₂×C₄×C₆₇	—	2	1	no	—
140	C₂×C₄×C₆₇	—	2	1	no	—
141	C₂×C₄×C₆₇	—	2	1	no	—
142	C₂×C₄×C₆₇	—	2	2	no	—
143	C₂×C₄×C₆₇	—	2	8	no	—
144	C₂×C₄×C₆₇	—	2	8	no	—
145	C₂×C₄×C₆₇	—	2	1	no	—
146	C₂×C₄×C₆₇	—	2	2	no	—
147	C₂×C₄×C₆₇	—	2	1	no	—
148	C₂×C₄×C₆₇	—	2	2	no	—
149	C₂×C₄×C₆₇	—	2	2	no	—
150	C₂×C₄×C₆₇	—	2	1	no	—
151	C₂×C₄×C₆₇	—	4	8	no	—
152	C₂×C₄×C₆₇	—	4	8	no	—
153	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
154	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
155	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
156	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
157	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
158	C₂×C₄×C₆₇	C₂²×C₈	1	4	yes	—
159	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
160	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
161	C₂×C₄×C₆₇	#131	1	2	yes	—
162	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
163	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
164	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
165	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
166	C₈×C₆₇	—	2	4	no	—
167	C₈×C₆₇	—	4	2	no	—
168	C₈×C₆₇	—	4	4	no	—
169	C₂²×C₆₇	—	2	4	no	—
170	C₂²×C₆₇	—	2	4	no	—
171	C₂²×C₆₇	—	2	4	no	—
172	C₂²×C₆₇	—	2	4	no	—
173	C₂²×C₆₇	—	2	4	no	—
174	C₂²×C₆₇	—	2	4	no	—
175	C₂²×C₆₇	C₂×C₄⋊C₈	1	8	yes	—
176	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
177	C₂²×C₆₇	C₄×#41	1	1	yes	characteristic
178	C₂²×C₆₇	#518	1	1	yes	characteristic
179	C₂²×C₆₇	C₂³×C₈	1	1	yes	characteristic
180	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
181	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
182	C₂²×C₆₇	#568	1	1	yes	characteristic
183	C₄×C₆₇	—	2	4	no	—
184	C₄×C₆₇	—	2	2	no	—
185	C₄×C₆₇	—	2	2	no	—
186	C₄×C₆₇	—	2	2	no	—
187	C₄×C₆₇	—	2	4	no	—
188	C₄×C₆₇	—	2	2	no	—
189	C₄×C₆₇	—	4	1	no	—
190	C₄×C₆₇	—	4	2	no	—
191	C₄×C₆₇	C₂×C₄⋊C₈	1	4	yes	—
192	C₄×C₆₇	#546	1	2	yes	—
193	C₄×C₆₇	#554	1	2	yes	—
194	#22696	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
195	C₂×C₆₇	—	2	8	no	—
196	C₂×C₆₇	#2349	1	1	yes	characteristic
197	C₂×C₆₇	C₂²×C₄⋊C₈	1	1	yes	characteristic
198	C₂×C₆₇	#2906	1	1	yes	characteristic
199	C₂×C₆₇	C₈×#41	1	1	yes	characteristic
200	C₂×C₆₇	#2970	1	1	yes	characteristic
201	C₂×C₆₇	#2972	1	1	yes	characteristic
202	C₂×C₆₇	#2974	1	1	yes	characteristic
203	C₂×C₈×D₄	C₂×C₆₇	1	2	yes	—
204	C₄²×D₄	C₂×C₆₇	1	1	yes	characteristic
205	#1939	C₂×C₆₇	1	2	yes	complemented
206	#1850	C₂×C₆₇	1	4	yes	complemented
207	#1831	C₂×C₆₇	1	1	yes	characteristic, complemented
208	#1797	C₂×C₆₇	1	2	yes	complemented
209	C₄×C₄⋊C₈	C₂×C₆₇	1	1	yes	characteristic, complemented
210	C₂×#336	C₂×C₆₇	1	2	yes	complemented
211	C₆₇	#22696	1	1	yes	characteristic, direct product factor, Sylow
212	C₂³×C₈	C₂²×C₆₇	1	4	yes	—
213	C₂×C₄×D₄	C₄×C₆₇	1	1	yes	characteristic
214	C₂×C₄×D₄	C₄×C₆₇	1	1	yes	characteristic
215	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
216	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
217	C₈×D₄	—	2	8	no	—
218	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
219	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
220	C₂×C₄⋊C₈	C₂²×C₆₇	1	4	yes	—
221	C₂×C₂²⋊C₈	C₂²×C₆₇	1	4	yes	—
222	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
223	C₄×C₄⋊C₄	C₄×C₆₇	1	1	yes	characteristic
224	C₄×C₄⋊C₄	C₄×C₆₇	1	1	yes	characteristic
225	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
226	C₄×C₂²⋊C₄	C₄×C₆₇	1	2	yes	—
227	C₄×C₂²⋊C₄	C₄×C₆₇	1	2	yes	—
228	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
229	C₄³	C₂²×C₆₇	1	1	yes	characteristic
230	#336	C₂²×C₆₇	1	1	yes	characteristic
231	#336	C₂²×C₆₇	1	2	yes	—
232	#336	C₂²×C₆₇	1	2	yes	—
233	#336	C₂²×C₆₇	1	4	yes	—
234	#336	C₂²×C₆₇	1	1	yes	characteristic
235	C₂²×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
236	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
237	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
238	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
239	C₂²×C₈	—	2	8	no	—
240	C₂²×C₈	—	2	8	no	—
241	C₂²×C₈	—	4	4	no	—
242	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
243	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
244	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
245	C₄×D₄	—	2	4	no	—
246	C₄×D₄	C₈×C₆₇	1	8	yes	complemented
247	C₄×D₄	D₄×C₆₇	1	4	yes	—
248	C₄×D₄	Q₈×C₆₇	1	4	yes	—
249	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
250	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
251	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
252	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
253	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
254	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
255	C₂×C₄²	—	4	4	no	—
256	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
257	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
258	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
259	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
260	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
261	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
262	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
263	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
264	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
265	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
266	C₄⋊C₈	—	2	4	no	—
267	C₄⋊C₈	—	2	4	no	—
268	C₄⋊C₈	—	4	4	no	—
269	C₂²⋊C₈	—	2	8	no	—
270	C₄×C₈	—	2	4	no	—
271	#95	C₂×C₄×C₆₇	1	2	yes	—
272	#95	C₂×C₄×C₆₇	1	2	yes	—
273	#95	C₂×C₄×C₆₇	1	2	yes	—
274	C₂⁴	C₂²×C₄×C₆₇	1	2	yes	—
275	C₂×D₄	C₄⋊C₄×C₆₇	1	4	yes	—
276	C₂×D₄	C₂×C₈×C₆₇	1	4	yes	—
277	C₂×D₄	M₁₆×C₆₇	1	4	yes	—
278	C₂²×C₄	—	2	4	no	—
279	C₂²×C₄	—	2	4	no	—
280	C₂²×C₄	—	2	4	no	—
281	C₂²×C₄	—	2	4	no	—
282	C₂²×C₄	—	4	2	no	—
283	C₂²×C₄	—	4	2	no	—
284	C₂²×C₄	—	4	2	no	—
285	C₂²×C₄	C₂×C₈×C₆₇	1	8	yes	complemented
286	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
287	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
288	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
289	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
290	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
291	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
292	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
293	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
294	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
295	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
296	C₂²×C₄	C₂×Q₈×C₆₇	1	4	yes	—
297	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
298	C₂×C₈	—	2	2	no	—
299	C₂×C₈	—	2	2	no	—
300	C₂×C₈	—	2	2	no	—
301	C₂×C₈	—	2	4	no	—
302	C₂×C₈	—	2	2	no	—
303	C₂×C₈	—	2	4	no	—
304	C₂×C₈	—	2	2	no	—
305	C₂×C₈	—	2	4	no	—
306	C₂×C₈	—	4	16	no	—
307	C₂×C₈	#41×C₆₇	1	4	yes	—
308	C₄⋊C₄	—	2	2	no	—
309	C₄⋊C₄	C₄⋊C₄×C₆₇	1	4	yes	—
310	C₄⋊C₄	C₂×C₈×C₆₇	1	4	yes	complemented
311	C₂²⋊C₄	—	2	4	no	—
312	C₂²⋊C₄	C₄⋊C₄×C₆₇	1	8	yes	—
313	C₂²⋊C₄	C₂×C₈×C₆₇	1	8	yes	complemented
314	C₄²	—	2	2	no	—
315	C₄²	—	2	2	no	—
316	C₄²	—	2	2	no	—
317	C₄²	—	2	4	no	—
318	C₄²	—	2	2	no	—
319	C₄²	—	4	2	no	—
320	C₄²	—	4	2	no	—
321	C₄²	—	4	1	no	—
322	C₄²	C₂×C₈×C₆₇	1	4	yes	complemented
323	C₄²	C₂×D₄×C₆₇	1	2	yes	—
324	C₄²	C₂×Q₈×C₆₇	1	2	yes	—
325	C₂³	—	2	2	no	—
326	C₂³	—	2	2	no	—
327	C₂³	—	2	2	no	—
328	C₂³	—	2	2	no	—
329	C₂³	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
330	C₂³	C₂²×C₈×C₆₇	1	4	yes	—
331	C₂³	C₂×M₁₆×C₆₇	1	4	yes	—
332	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
333	D₄	C₄⋊C₈×C₆₇	1	16	yes	complemented
334	C₂×C₄	—	2	1	no	—
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	2	1	no	—
337	C₂×C₄	—	2	1	no	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	2	2	no	—
340	C₂×C₄	—	2	1	no	—
341	C₂×C₄	—	2	1	no	—
342	C₂×C₄	—	2	1	no	—
343	C₂×C₄	—	2	1	no	—
344	C₂×C₄	—	2	1	no	—
345	C₂×C₄	—	2	2	no	—
346	C₂×C₄	—	2	8	no	—
347	C₂×C₄	—	2	8	no	—
348	C₂×C₄	—	2	1	no	—
349	C₂×C₄	—	2	2	no	—
350	C₂×C₄	—	2	1	no	—
351	C₂×C₄	—	2	2	no	—
352	C₂×C₄	—	2	2	no	—
353	C₂×C₄	—	2	1	no	—
354	C₂×C₄	—	4	8	no	—
355	C₂×C₄	—	4	8	no	—
356	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
357	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
358	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
359	C₂×C₄	C₂²×C₈×C₆₇	1	2	yes	—
360	C₂×C₄	C₂²×C₈×C₆₇	1	2	yes	—
361	C₂×C₄	C₂²×C₈×C₆₇	1	4	yes	—
362	C₂×C₄	C₂²×C₈×C₆₇	1	2	yes	—
363	C₂×C₄	C₂×M₁₆×C₆₇	1	2	yes	—
364	C₂×C₄	#131×C₆₇	1	2	yes	—
365	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
366	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
367	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
368	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
369	C₈	—	2	4	no	—
370	C₈	—	4	4	no	—
371	C₈	—	4	2	no	—
372	C₂²	—	2	4	no	—
373	C₂²	—	2	4	no	—
374	C₂²	—	2	4	no	—
375	C₂²	—	2	4	no	—
376	C₂²	—	2	4	no	—
377	C₂²	—	2	4	no	—
378	C₂²	C₂×C₄⋊C₈×C₆₇	1	8	yes	—
379	C₂²	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
380	C₂²	C₄×#41×C₆₇	1	1	yes	characteristic
381	C₂²	#518×C₆₇	1	1	yes	characteristic
382	C₂²	C₂³×C₈×C₆₇	1	1	yes	characteristic
383	C₂²	C₂²×M₁₆×C₆₇	1	1	yes	characteristic
384	C₂²	C₂×#131×C₆₇	1	1	yes	characteristic
385	C₂²	#568×C₆₇	1	1	yes	characteristic
386	C₄	—	2	4	no	—
387	C₄	—	2	2	no	—
388	C₄	—	2	2	no	—
389	C₄	—	2	2	no	—
390	C₄	—	2	4	no	—
391	C₄	—	2	2	no	—
392	C₄	—	4	1	no	—
393	C₄	—	4	2	no	—
394	C₄	C₂×C₄⋊C₈×C₆₇	1	4	yes	—
395	C₄	#546×C₆₇	1	2	yes	—
396	C₄	#554×C₆₇	1	2	yes	—
397	C₂	—	2	8	no	—
398	C₂	C₆₇×#2349	1	1	yes	characteristic
399	C₂	C₂²×C₄⋊C₈×C₆₇	1	1	yes	characteristic
400	C₂	C₆₇×#2906	1	1	yes	characteristic
401	C₂	C₈×#41×C₆₇	1	1	yes	characteristic
402	C₂	C₆₇×#2970	1	1	yes	characteristic
403	C₂	C₆₇×#2972	1	1	yes	characteristic
404	C₂	C₆₇×#2974	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂⁴
commutator	C₂²	C₂³×C₈×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂³×C₆₇	C₂³×C₄
prime radical	C₂²×D₄×C₆₇	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₆₇×#1939	C₂	semidirect
4. C₆₇×#1850	C₂	semidirect
5. C₆₇×#1831	C₂	semidirect
6. C₆₇×#1797	C₂	semidirect
7. C₄×C₄⋊C₈×C₆₇	C₂	semidirect
8. C₂×#336×C₆₇	C₂	semidirect
43. C₄×D₄×C₆₇	C₈	semidirect
82. C₂²×C₄×C₆₇	C₂×C₈	semidirect
107. C₄⋊C₄×C₆₇	C₂×C₈	semidirect
110. C₂²⋊C₄×C₆₇	C₂×C₈	semidirect
119. C₄²×C₆₇	C₂×C₈	semidirect
130. D₄×C₆₇	C₄⋊C₈	semidirect
194. #22696	C₆₇	direct (N × H)
205. #1939	C₂×C₆₇	semidirect
206. #1850	C₂×C₆₇	semidirect
207. #1831	C₂×C₆₇	semidirect
208. #1797	C₂×C₆₇	semidirect
209. C₄×C₄⋊C₈	C₂×C₆₇	semidirect
210. C₂×#336	C₂×C₆₇	semidirect
211. C₆₇	#22696	direct (N × H)
246. C₄×D₄	C₈×C₆₇	semidirect
285. C₂²×C₄	C₂×C₈×C₆₇	semidirect
310. C₄⋊C₄	C₂×C₈×C₆₇	semidirect
313. C₂²⋊C₄	C₂×C₈×C₆₇	semidirect
322. C₄²	C₂×C₈×C₆₇	semidirect
333. D₄	C₄⋊C₈×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₄×C₆₇	2
C₄²×D₄×C₆₇	2
C₆₇×#1939	2
C₆₇×#1850	2
C₆₇×#1831	2
C₆₇×#1797	2
C₄×C₄⋊C₈×C₆₇	2
C₂×#336×C₆₇	2
#22696	67

Subgroups — C67×#22696 #30701947

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22696 #30701947