Subgroups — C₆₇×#20808 #30700059

Order 17152.

1174 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#431×C₆₇	C₂	1	2	yes	maximal
2	C₂×C₄³×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#1842	C₂	1	2	yes	maximal, complemented
4	C₆₇×#1831	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1797	C₂	1	4	yes	maximal
6	C₄×C₄⋊C₈×C₆₇	C₂	1	4	yes	maximal, complemented
7	C₂²×C₄²×C₆₇	C₄	1	1	yes	characteristic
8	C₂²×C₄²×C₆₇	C₄	1	2	yes	—
9	C₂²×C₄²×C₆₇	C₄	1	1	yes	characteristic
10	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
11	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
12	#431×C₆₇	—	2	8	no	—
13	C₂×C₄⋊C₈×C₆₇	C₂²	1	8	yes	—
14	C₂×C₄⋊C₈×C₆₇	C₂²	1	4	yes	—
15	C₂×C₂²⋊C₈×C₆₇	C₂²	1	4	yes	—
16	C₂×C₄×C₈×C₆₇	C₂²	1	4	yes	—
17	C₄³×C₆₇	C₄	1	2	yes	—
18	C₄³×C₆₇	C₄	1	1	yes	characteristic
19	C₄³×C₆₇	C₄	1	1	yes	characteristic
20	C₄³×C₆₇	C₂²	1	1	yes	characteristic
21	C₄³×C₆₇	C₂²	1	1	yes	characteristic
22	C₄³×C₆₇	C₂²	1	2	yes	—
23	#336×C₆₇	C₂²	1	8	yes	—
24	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
25	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
27	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
29	C₂²×C₈×C₆₇	C₂³	1	8	yes	—
30	C₂×C₄²×C₆₇	—	2	4	no	—
31	C₂×C₄²×C₆₇	—	2	8	no	—
32	C₂×C₄²×C₆₇	—	2	8	no	—
33	C₂×C₄²×C₆₇	C₈	1	8	yes	complemented
34	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₆₇	D₄	1	4	yes	—
49	C₂×C₄²×C₆₇	Q₈	1	4	yes	—
50	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
51	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
53	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
54	C₄⋊C₈×C₆₇	—	2	8	no	—
55	C₄⋊C₈×C₆₇	—	2	16	no	—
56	C₂²⋊C₈×C₆₇	—	2	8	no	—
57	C₄×C₈×C₆₇	—	2	8	no	—
58	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₆₇	—	2	4	no	—
60	C₂²×C₄×C₆₇	—	2	4	no	—
61	C₂²×C₄×C₆₇	—	2	4	no	—
62	C₂²×C₄×C₆₇	—	2	4	no	—
63	C₂²×C₄×C₆₇	—	2	4	no	—
64	C₂²×C₄×C₆₇	—	2	4	no	—
65	C₂²×C₄×C₆₇	—	2	4	no	—
66	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
67	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
68	C₂²×C₄×C₆₇	C₂×C₈	1	8	yes	—
69	C₂²×C₄×C₆₇	C₂×C₈	1	4	yes	—
70	C₂²×C₄×C₆₇	M₁₆	1	4	yes	—
71	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
82	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
83	C₂×C₈×C₆₇	—	2	8	no	—
84	C₂×C₈×C₆₇	—	2	16	no	—
85	C₄²×C₆₇	—	2	4	no	—
86	C₄²×C₆₇	—	2	4	no	—
87	C₄²×C₆₇	—	2	4	no	—
88	C₄²×C₆₇	—	2	2	no	—
89	C₄²×C₆₇	—	2	4	no	—
90	C₄²×C₆₇	—	2	2	no	—
91	C₄²×C₆₇	—	2	4	no	—
92	C₄²×C₆₇	—	2	4	no	—
93	C₄²×C₆₇	—	2	4	no	—
94	C₄²×C₆₇	C₄⋊C₄	1	4	yes	—
95	C₄²×C₆₇	C₄⋊C₄	1	4	yes	—
96	C₄²×C₆₇	C₂×C₈	1	4	yes	—
97	C₄²×C₆₇	C₂×C₈	1	8	yes	—
98	C₄²×C₆₇	C₂×C₈	1	4	yes	—
99	C₄²×C₆₇	M₁₆	1	8	yes	—
100	C₄²×C₆₇	C₂×D₄	1	2	yes	—
101	C₄²×C₆₇	C₂×D₄	1	2	yes	—
102	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
103	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
104	C₄²×C₆₇	#41	1	8	yes	—
105	C₂³×C₆₇	—	2	2	no	—
106	C₂³×C₆₇	—	2	2	no	—
107	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
108	C₂³×C₆₇	C₂²×C₈	1	2	yes	—
109	C₂³×C₆₇	C₂×M₁₆	1	2	yes	—
110	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
111	C₂×C₄×C₆₇	—	2	16	no	—
112	C₂×C₄×C₆₇	—	2	2	no	—
113	C₂×C₄×C₆₇	—	2	2	no	—
114	C₂×C₄×C₆₇	—	2	2	no	—
115	C₂×C₄×C₆₇	—	2	2	no	—
116	C₂×C₄×C₆₇	—	2	2	no	—
117	C₂×C₄×C₆₇	—	2	2	no	—
118	C₂×C₄×C₆₇	—	2	2	no	—
119	C₂×C₄×C₆₇	—	2	2	no	—
120	C₂×C₄×C₆₇	—	2	4	no	—
121	C₂×C₄×C₆₇	—	2	4	no	—
122	C₂×C₄×C₆₇	—	2	8	no	—
123	C₂×C₄×C₆₇	—	2	16	no	—
124	C₂×C₄×C₆₇	C₄⋊C₈	1	16	yes	complemented
125	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
126	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
127	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
128	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
129	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
130	C₂×C₄×C₆₇	#117	1	4	yes	—
131	C₂×C₄×C₆₇	#117	1	4	yes	—
132	C₂×C₄×C₆₇	C₂²×C₈	1	4	yes	—
133	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
134	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
135	C₂×C₄×C₆₇	C₂²×C₈	1	4	yes	—
136	C₂×C₄×C₆₇	C₂×M₁₆	1	4	yes	—
137	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
138	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
139	C₂×C₄×C₆₇	C₂×M₁₆	1	4	yes	—
140	C₂×C₄×C₆₇	C₂×M₁₆	1	4	yes	—
141	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
142	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
143	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
144	C₈×C₆₇	—	4	8	no	—
145	C₂²×C₆₇	—	2	2	no	—
146	C₂²×C₆₇	—	2	4	no	—
147	C₂²×C₆₇	—	2	4	no	—
148	C₂²×C₆₇	—	2	2	no	—
149	C₂²×C₆₇	C₂×C₄⋊C₈	1	4	yes	—
150	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
151	C₂²×C₆₇	C₂×#117	1	2	yes	—
152	C₂²×C₆₇	C₂³×C₈	1	1	yes	characteristic
153	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
154	C₂²×C₆₇	C₂²×M₁₆	1	2	yes	—
155	C₄×C₆₇	—	2	2	no	—
156	C₄×C₆₇	—	2	4	no	—
157	C₄×C₆₇	—	2	4	no	—
158	C₄×C₆₇	—	2	4	no	—
159	C₄×C₆₇	—	2	2	no	—
160	C₄×C₆₇	C₂×C₄⋊C₈	1	4	yes	—
161	C₄×C₆₇	C₂×C₄⋊C₈	1	4	yes	—
162	C₄×C₆₇	#423	1	4	yes	—
163	C₄×C₆₇	#431	1	8	yes	complemented
164	C₄×C₆₇	#532	1	2	yes	—
165	C₄×C₆₇	#533	1	2	yes	—
166	#20808	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
167	C₂×C₆₇	—	2	4	no	—
168	C₂×C₆₇	#2285	1	1	yes	characteristic
169	C₂×C₆₇	C₂²×C₄⋊C₈	1	1	yes	characteristic
170	C₂×C₆₇	C₂×#423	1	1	yes	characteristic
171	C₂×C₆₇	C₂×#431	1	2	yes	—
172	C₂×C₆₇	#2920	1	2	yes	—
173	C₂×#431	C₂×C₆₇	1	2	yes	—
174	C₂×C₄³	C₂×C₆₇	1	1	yes	characteristic
175	#1842	C₂×C₆₇	1	2	yes	complemented
176	#1831	C₂×C₆₇	1	2	yes	complemented
177	#1797	C₂×C₆₇	1	4	yes	—
178	C₄×C₄⋊C₈	C₂×C₆₇	1	4	yes	complemented
179	C₆₇	#20808	1	1	yes	characteristic, direct product factor, Sylow
180	C₂²×C₄²	C₄×C₆₇	1	1	yes	characteristic
181	C₂²×C₄²	C₄×C₆₇	1	2	yes	—
182	C₂²×C₄²	C₄×C₆₇	1	1	yes	characteristic
183	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
184	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
185	#431	—	2	8	no	—
186	C₂×C₄⋊C₈	C₂²×C₆₇	1	8	yes	—
187	C₂×C₄⋊C₈	C₂²×C₆₇	1	4	yes	—
188	C₂×C₂²⋊C₈	C₂²×C₆₇	1	4	yes	—
189	C₂×C₄×C₈	C₂²×C₆₇	1	4	yes	—
190	C₄³	C₄×C₆₇	1	1	yes	characteristic
191	C₄³	C₄×C₆₇	1	2	yes	—
192	C₄³	C₄×C₆₇	1	1	yes	characteristic
193	C₄³	C₂²×C₆₇	1	1	yes	characteristic
194	C₄³	C₂²×C₆₇	1	1	yes	characteristic
195	C₄³	C₂²×C₆₇	1	2	yes	—
196	#336	C₂²×C₆₇	1	8	yes	—
197	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
198	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
199	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
200	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
201	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
202	C₂²×C₈	C₂³×C₆₇	1	8	yes	—
203	C₂×C₄²	—	2	4	no	—
204	C₂×C₄²	—	2	8	no	—
205	C₂×C₄²	—	2	8	no	—
206	C₂×C₄²	C₈×C₆₇	1	8	yes	complemented
207	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
208	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
209	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
210	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
211	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
212	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
213	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
214	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
215	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
216	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
217	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
218	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
219	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
220	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
221	C₂×C₄²	D₄×C₆₇	1	4	yes	—
222	C₂×C₄²	Q₈×C₆₇	1	4	yes	—
223	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
224	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
225	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
226	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
227	C₄⋊C₈	—	2	8	no	—
228	C₄⋊C₈	—	2	16	no	—
229	C₂²⋊C₈	—	2	8	no	—
230	C₄×C₈	—	2	8	no	—
231	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
232	C₂²×C₄	—	2	4	no	—
233	C₂²×C₄	—	2	4	no	—
234	C₂²×C₄	—	2	4	no	—
235	C₂²×C₄	—	2	4	no	—
236	C₂²×C₄	—	2	4	no	—
237	C₂²×C₄	—	2	4	no	—
238	C₂²×C₄	—	2	4	no	—
239	C₂²×C₄	C₄⋊C₄×C₆₇	1	4	yes	—
240	C₂²×C₄	C₄⋊C₄×C₆₇	1	4	yes	—
241	C₂²×C₄	C₂×C₈×C₆₇	1	8	yes	—
242	C₂²×C₄	C₂×C₈×C₆₇	1	4	yes	—
243	C₂²×C₄	M₁₆×C₆₇	1	4	yes	—
244	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
245	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
246	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
247	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
248	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
249	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
250	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
251	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
252	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
253	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
254	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
255	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
256	C₂×C₈	—	2	8	no	—
257	C₂×C₈	—	2	16	no	—
258	C₄²	—	2	4	no	—
259	C₄²	—	2	4	no	—
260	C₄²	—	2	4	no	—
261	C₄²	—	2	2	no	—
262	C₄²	—	2	4	no	—
263	C₄²	—	2	2	no	—
264	C₄²	—	2	4	no	—
265	C₄²	—	2	4	no	—
266	C₄²	—	2	4	no	—
267	C₄²	C₄⋊C₄×C₆₇	1	4	yes	—
268	C₄²	C₄⋊C₄×C₆₇	1	4	yes	—
269	C₄²	C₂×C₈×C₆₇	1	4	yes	—
270	C₄²	C₂×C₈×C₆₇	1	8	yes	—
271	C₄²	C₂×C₈×C₆₇	1	4	yes	—
272	C₄²	M₁₆×C₆₇	1	8	yes	—
273	C₄²	C₂×D₄×C₆₇	1	2	yes	—
274	C₄²	C₂×D₄×C₆₇	1	2	yes	—
275	C₄²	C₂×Q₈×C₆₇	1	2	yes	—
276	C₄²	C₂×Q₈×C₆₇	1	2	yes	—
277	C₄²	#41×C₆₇	1	8	yes	—
278	C₂³	—	2	2	no	—
279	C₂³	—	2	2	no	—
280	C₂³	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
281	C₂³	C₂²×C₈×C₆₇	1	2	yes	—
282	C₂³	C₂×M₁₆×C₆₇	1	2	yes	—
283	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
284	C₂×C₄	—	2	16	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	4	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	8	no	—
296	C₂×C₄	—	2	16	no	—
297	C₂×C₄	C₄⋊C₈×C₆₇	1	16	yes	complemented
298	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
299	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
300	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
301	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
302	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
303	C₂×C₄	#117×C₆₇	1	4	yes	—
304	C₂×C₄	#117×C₆₇	1	4	yes	—
305	C₂×C₄	C₂²×C₈×C₆₇	1	4	yes	—
306	C₂×C₄	C₂²×C₈×C₆₇	1	2	yes	—
307	C₂×C₄	C₂²×C₈×C₆₇	1	2	yes	—
308	C₂×C₄	C₂²×C₈×C₆₇	1	4	yes	—
309	C₂×C₄	C₂×M₁₆×C₆₇	1	4	yes	—
310	C₂×C₄	C₂×M₁₆×C₆₇	1	2	yes	—
311	C₂×C₄	C₂×M₁₆×C₆₇	1	2	yes	—
312	C₂×C₄	C₂×M₁₆×C₆₇	1	4	yes	—
313	C₂×C₄	C₂×M₁₆×C₆₇	1	4	yes	—
314	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
315	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
316	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
317	C₈	—	4	8	no	—
318	C₂²	—	2	2	no	—
319	C₂²	—	2	2	no	—
320	C₂²	—	2	4	no	—
321	C₂²	—	2	4	no	—
322	C₂²	C₂×C₄⋊C₈×C₆₇	1	4	yes	—
323	C₂²	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
324	C₂²	C₂×#117×C₆₇	1	2	yes	—
325	C₂²	C₂³×C₈×C₆₇	1	1	yes	characteristic
326	C₂²	C₂²×M₁₆×C₆₇	1	1	yes	characteristic
327	C₂²	C₂²×M₁₆×C₆₇	1	2	yes	—
328	C₄	—	2	2	no	—
329	C₄	—	2	2	no	—
330	C₄	—	2	4	no	—
331	C₄	—	2	4	no	—
332	C₄	—	2	4	no	—
333	C₄	C₂×C₄⋊C₈×C₆₇	1	4	yes	—
334	C₄	C₂×C₄⋊C₈×C₆₇	1	4	yes	—
335	C₄	#423×C₆₇	1	4	yes	—
336	C₄	#431×C₆₇	1	8	yes	complemented
337	C₄	#532×C₆₇	1	2	yes	—
338	C₄	#533×C₆₇	1	2	yes	—
339	C₂	—	2	4	no	—
340	C₂	C₆₇×#2285	1	1	yes	characteristic
341	C₂	C₂²×C₄⋊C₈×C₆₇	1	1	yes	characteristic
342	C₂	C₂×#423×C₆₇	1	1	yes	characteristic
343	C₂	C₂×#431×C₆₇	1	2	yes	—
344	C₂	C₆₇×#2920	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²×C₆₇	C₂³
commutator	C₂²	C₂³×C₈×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂³×C₆₇	C₂³×C₄
prime radical	C₂⁴×C₆₇	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₆₇×#1842	C₂	semidirect
4. C₆₇×#1831	C₂	semidirect
6. C₄×C₄⋊C₈×C₆₇	C₂	semidirect
33. C₂×C₄²×C₆₇	C₈	semidirect
124. C₂×C₄×C₆₇	C₄⋊C₈	semidirect
163. C₄×C₆₇	#431	semidirect
166. #20808	C₆₇	direct (N × H)
175. #1842	C₂×C₆₇	semidirect
176. #1831	C₂×C₆₇	semidirect
178. C₄×C₄⋊C₈	C₂×C₆₇	semidirect
179. C₆₇	#20808	direct (N × H)
206. C₂×C₄²	C₈×C₆₇	semidirect
297. C₂×C₄	C₄⋊C₈×C₆₇	semidirect
336. C₄	#431×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#431×C₆₇	2
C₂×C₄³×C₆₇	2
C₆₇×#1842	2
C₆₇×#1831	2
C₆₇×#1797	2
C₄×C₄⋊C₈×C₆₇	2
#20808	67

Subgroups — C67×#20808 #30700059

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#20808 #30700059