Subgroups — C₆₇×#20389 #30699640

Order 17152.

1670 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₂²⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#405×C₆₇	C₂	1	2	yes	maximal, complemented
4	C₆₇×#2280	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1753	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1752	C₂	1	2	yes	maximal, complemented
7	C₈×C₂²⋊C₄×C₆₇	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1729	C₂	1	2	yes	maximal, complemented
9	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
10	C₂²×M₁₆×C₆₇	—	2	8	no	—
11	C₂²×M₁₆×C₆₇	C₄	1	8	yes	complemented
12	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
13	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
14	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
16	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
17	C₂×#117×C₆₇	C₂²	1	2	yes	—
18	C₂×C₂²⋊C₈×C₆₇	C₄	1	8	yes	complemented
19	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
22	#405×C₆₇	—	2	8	no	—
23	C₂×#97×C₆₇	C₂²	1	4	yes	complemented
24	C₂×C₄×C₈×C₆₇	C₂²	1	4	yes	complemented
25	#381×C₆₇	C₄	1	2	yes	—
26	#380×C₆₇	C₄	1	2	yes	—
27	#380×C₆₇	C₄	1	2	yes	—
28	#379×C₆₇	C₄	1	2	yes	—
29	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
30	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
31	#336×C₆₇	C₂²	1	4	yes	complemented
32	#336×C₆₇	C₂²	1	4	yes	complemented
33	C₂⁵×C₆₇	C₂×C₄	1	1	yes	characteristic
34	C₂³×C₄×C₆₇	—	2	4	no	—
35	C₂³×C₄×C₆₇	—	2	4	no	—
36	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
37	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
38	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
39	C₂³×C₄×C₆₇	C₂×C₄	1	4	yes	—
40	C₂³×C₄×C₆₇	C₂×C₄	1	4	yes	—
41	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
42	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
43	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
44	C₂×M₁₆×C₆₇	—	2	4	no	—
45	C₂×M₁₆×C₆₇	—	2	32	no	—
46	C₂×M₁₆×C₆₇	—	2	8	no	—
47	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
48	C₂×M₁₆×C₆₇	C₂×C₄	1	8	yes	—
49	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
50	C₂×M₁₆×C₆₇	D₄	1	8	yes	—
51	C₂²×C₈×C₆₇	—	2	4	no	—
52	C₂²×C₈×C₆₇	—	2	4	no	—
53	C₂²×C₈×C₆₇	—	2	8	no	—
54	C₂²×C₈×C₆₇	C₂×C₄	1	8	yes	—
55	C₂²×C₈×C₆₇	C₂×C₄	1	8	yes	—
56	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
57	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
58	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
59	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
60	#117×C₆₇	—	2	4	no	—
61	#117×C₆₇	—	2	4	no	—
62	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
63	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
64	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
65	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
66	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
67	C₂²⋊C₈×C₆₇	C₂×C₄	1	4	yes	—
68	C₂²⋊C₈×C₆₇	C₂×C₄	1	4	yes	—
69	C₂²⋊C₈×C₆₇	C₂×C₄	1	8	yes	—
70	#97×C₆₇	—	2	8	no	—
71	C₄×C₈×C₆₇	—	2	8	no	—
72	#95×C₆₇	C₂×C₄	1	4	yes	—
73	#95×C₆₇	C₂×C₄	1	4	yes	—
74	C₂⁴×C₆₇	—	2	4	no	—
75	C₂⁴×C₆₇	—	2	4	no	—
76	C₂⁴×C₆₇	—	4	2	no	—
77	C₂⁴×C₆₇	C₄²	1	4	yes	—
78	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
79	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
80	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₆₇	—	2	4	no	—
82	C₂²×C₄×C₆₇	—	2	8	no	—
83	C₂²×C₄×C₆₇	—	2	8	no	—
84	C₂²×C₄×C₆₇	—	2	4	no	—
85	C₂²×C₄×C₆₇	—	2	4	no	—
86	C₂²×C₄×C₆₇	—	2	16	no	—
87	C₂²×C₄×C₆₇	—	2	2	no	—
88	C₂²×C₄×C₆₇	—	2	2	no	—
89	C₂²×C₄×C₆₇	—	2	2	no	—
90	C₂²×C₄×C₆₇	—	2	2	no	—
91	C₂²×C₄×C₆₇	C₄²	1	4	yes	—
92	C₂²×C₄×C₆₇	C₄²	1	4	yes	—
93	C₂²×C₄×C₆₇	C₄²	1	4	yes	—
94	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
95	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
96	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
98	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
99	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
100	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
101	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
102	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
103	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
104	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
105	C₂²×C₄×C₆₇	#41	1	4	yes	—
106	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
107	M₁₆×C₆₇	—	2	8	no	—
108	M₁₆×C₆₇	—	2	16	no	—
109	M₁₆×C₆₇	C₂²⋊C₄	1	16	yes	complemented
110	C₂×C₈×C₆₇	—	2	16	no	—
111	C₂×C₈×C₆₇	—	2	4	no	—
112	C₂×C₈×C₆₇	—	2	2	no	—
113	C₂×C₈×C₆₇	—	2	2	no	—
114	C₂×C₈×C₆₇	—	2	2	no	—
115	C₂×C₈×C₆₇	—	2	2	no	—
116	C₂×C₈×C₆₇	—	2	16	no	—
117	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
118	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
119	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
120	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
121	C₂×C₈×C₆₇	C₂×D₄	1	8	yes	—
122	C₄⋊C₄×C₆₇	—	2	8	no	—
123	C₂²⋊C₄×C₆₇	—	2	8	no	—
124	C₂²⋊C₄×C₆₇	—	2	8	no	—
125	C₂²⋊C₄×C₆₇	—	4	4	no	—
126	C₄²×C₆₇	—	2	4	no	—
127	C₄²×C₆₇	—	2	4	no	—
128	C₂³×C₆₇	—	2	2	no	—
129	C₂³×C₆₇	—	2	2	no	—
130	C₂³×C₆₇	—	2	2	no	—
131	C₂³×C₆₇	—	2	2	no	—
132	C₂³×C₆₇	—	2	2	no	—
133	C₂³×C₆₇	—	2	2	no	—
134	C₂³×C₆₇	—	2	2	no	—
135	C₂³×C₆₇	—	2	8	no	—
136	C₂³×C₆₇	—	2	8	no	—
137	C₂³×C₆₇	—	4	4	no	—
138	C₂³×C₆₇	—	4	16	no	—
139	C₂³×C₆₇	C₂×C₄²	1	2	yes	—
140	C₂³×C₆₇	C₂×C₄²	1	2	yes	—
141	C₂³×C₆₇	C₂×C₄²	1	2	yes	—
142	C₂³×C₆₇	C₄×D₄	1	4	yes	—
143	C₂³×C₆₇	#131	1	4	yes	—
144	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
145	C₂×C₄×C₆₇	—	2	2	no	—
146	C₂×C₄×C₆₇	—	2	2	no	—
147	C₂×C₄×C₆₇	—	2	8	no	—
148	C₂×C₄×C₆₇	—	2	2	no	—
149	C₂×C₄×C₆₇	—	2	2	no	—
150	C₂×C₄×C₆₇	—	2	2	no	—
151	C₂×C₄×C₆₇	—	2	2	no	—
152	C₂×C₄×C₆₇	—	2	4	no	—
153	C₂×C₄×C₆₇	—	2	4	no	—
154	C₂×C₄×C₆₇	—	2	8	no	—
155	C₂×C₄×C₆₇	—	2	8	no	—
156	C₂×C₄×C₆₇	—	2	8	no	—
157	C₂×C₄×C₆₇	—	2	8	no	—
158	C₂×C₄×C₆₇	—	2	8	no	—
159	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
160	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
161	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
162	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
163	C₂×C₄×C₆₇	#117	1	4	yes	—
164	C₂×C₄×C₆₇	C₄×D₄	1	8	yes	—
165	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
166	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
167	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
168	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
169	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
170	C₈×C₆₇	—	2	4	no	—
171	C₈×C₆₇	—	2	4	no	—
172	C₈×C₆₇	—	2	4	no	—
173	C₈×C₆₇	C₂×C₂²⋊C₄	1	8	yes	—
174	C₂²×C₆₇	—	2	2	no	—
175	C₂²×C₆₇	—	2	2	no	—
176	C₂²×C₆₇	—	2	2	no	—
177	C₂²×C₆₇	—	2	8	no	—
178	C₂²×C₆₇	—	2	8	no	—
179	C₂²×C₆₇	—	2	2	no	—
180	C₂²×C₆₇	—	2	2	no	—
181	C₂²×C₆₇	—	2	2	no	—
182	C₂²×C₆₇	—	2	2	no	—
183	C₂²×C₆₇	—	4	16	no	—
184	C₂²×C₆₇	—	4	4	no	—
185	C₂²×C₆₇	C₄×C₂²⋊C₄	1	4	yes	—
186	C₂²×C₆₇	#405	1	4	yes	complemented
187	C₂²×C₆₇	C₂²×C₄²	1	1	yes	characteristic
188	C₂²×C₆₇	C₂×C₄×D₄	1	2	yes	—
189	C₂²×C₆₇	C₂×#131	1	2	yes	—
190	C₂²×C₆₇	#568	1	2	yes	—
191	C₄×C₆₇	—	2	2	no	—
192	C₄×C₆₇	—	2	2	no	—
193	C₄×C₆₇	—	4	8	no	—
194	C₄×C₆₇	C₄×C₂²⋊C₄	1	4	yes	—
195	C₄×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
196	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
197	C₄×C₆₇	C₂×C₄×D₄	1	2	yes	—
198	#20389	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
199	C₂×C₆₇	—	2	4	no	—
200	C₂×C₆₇	—	2	4	no	—
201	C₂×C₆₇	—	4	2	no	—
202	C₂×C₆₇	C₂×C₄×C₂²⋊C₄	1	1	yes	characteristic
203	C₂×C₆₇	C₂×#405	1	1	yes	characteristic
204	C₂×C₆₇	#2873	1	1	yes	characteristic
205	C₂×C₆₇	#2933	1	4	yes	—
206	C₂³×M₁₆	C₂×C₆₇	1	1	yes	characteristic
207	C₂²×C₂²⋊C₈	C₂×C₆₇	1	1	yes	characteristic
208	C₂×#405	C₂×C₆₇	1	2	yes	complemented
209	#2280	C₂×C₆₇	1	1	yes	characteristic
210	#1753	C₂×C₆₇	1	4	yes	complemented
211	#1752	C₂×C₆₇	1	2	yes	complemented
212	C₈×C₂²⋊C₄	C₂×C₆₇	1	2	yes	complemented
213	#1729	C₂×C₆₇	1	2	yes	complemented
214	C₆₇	#20389	1	1	yes	characteristic, direct product factor, Sylow
215	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
216	C₂²×M₁₆	—	2	8	no	—
217	C₂²×M₁₆	C₄×C₆₇	1	8	yes	complemented
218	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
219	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
220	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
221	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
222	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
223	C₂×#117	C₂²×C₆₇	1	2	yes	—
224	C₂×C₂²⋊C₈	C₄×C₆₇	1	8	yes	complemented
225	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
226	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
227	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
228	#405	—	2	8	no	—
229	C₂×#97	C₂²×C₆₇	1	4	yes	complemented
230	C₂×C₄×C₈	C₂²×C₆₇	1	4	yes	complemented
231	#381	C₄×C₆₇	1	2	yes	—
232	#380	C₄×C₆₇	1	2	yes	—
233	#380	C₄×C₆₇	1	2	yes	—
234	#379	C₄×C₆₇	1	2	yes	—
235	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
236	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
237	#336	C₂²×C₆₇	1	4	yes	complemented
238	#336	C₂²×C₆₇	1	4	yes	complemented
239	C₂⁵	C₂×C₄×C₆₇	1	1	yes	characteristic
240	C₂³×C₄	—	2	4	no	—
241	C₂³×C₄	—	2	4	no	—
242	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
243	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
244	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
245	C₂³×C₄	C₂×C₄×C₆₇	1	4	yes	—
246	C₂³×C₄	C₂×C₄×C₆₇	1	4	yes	—
247	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
248	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
249	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
250	C₂×M₁₆	—	2	4	no	—
251	C₂×M₁₆	—	2	32	no	—
252	C₂×M₁₆	—	2	8	no	—
253	C₂×M₁₆	C₂×C₄×C₆₇	1	4	yes	—
254	C₂×M₁₆	C₂×C₄×C₆₇	1	8	yes	—
255	C₂×M₁₆	C₂×C₄×C₆₇	1	4	yes	—
256	C₂×M₁₆	D₄×C₆₇	1	8	yes	—
257	C₂²×C₈	—	2	4	no	—
258	C₂²×C₈	—	2	4	no	—
259	C₂²×C₈	—	2	8	no	—
260	C₂²×C₈	C₂×C₄×C₆₇	1	8	yes	—
261	C₂²×C₈	C₂×C₄×C₆₇	1	8	yes	—
262	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
263	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
264	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
265	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
266	#117	—	2	4	no	—
267	#117	—	2	4	no	—
268	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
269	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
270	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
271	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
272	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
273	C₂²⋊C₈	C₂×C₄×C₆₇	1	4	yes	—
274	C₂²⋊C₈	C₂×C₄×C₆₇	1	4	yes	—
275	C₂²⋊C₈	C₂×C₄×C₆₇	1	8	yes	—
276	#97	—	2	8	no	—
277	C₄×C₈	—	2	8	no	—
278	#95	C₂×C₄×C₆₇	1	4	yes	—
279	#95	C₂×C₄×C₆₇	1	4	yes	—
280	C₂⁴	—	2	4	no	—
281	C₂⁴	—	2	4	no	—
282	C₂⁴	—	4	2	no	—
283	C₂⁴	C₄²×C₆₇	1	4	yes	—
284	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
285	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
286	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
287	C₂²×C₄	—	2	4	no	—
288	C₂²×C₄	—	2	8	no	—
289	C₂²×C₄	—	2	8	no	—
290	C₂²×C₄	—	2	4	no	—
291	C₂²×C₄	—	2	4	no	—
292	C₂²×C₄	—	2	16	no	—
293	C₂²×C₄	—	2	2	no	—
294	C₂²×C₄	—	2	2	no	—
295	C₂²×C₄	—	2	2	no	—
296	C₂²×C₄	—	2	2	no	—
297	C₂²×C₄	C₄²×C₆₇	1	4	yes	—
298	C₂²×C₄	C₄²×C₆₇	1	4	yes	—
299	C₂²×C₄	C₄²×C₆₇	1	4	yes	—
300	C₂²×C₄	C₂²×C₄×C₆₇	1	8	yes	—
301	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
302	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
303	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
304	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
305	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
306	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
307	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
308	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
309	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
310	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
311	C₂²×C₄	#41×C₆₇	1	4	yes	—
312	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
313	M₁₆	—	2	8	no	—
314	M₁₆	—	2	16	no	—
315	M₁₆	C₂²⋊C₄×C₆₇	1	16	yes	complemented
316	C₂×C₈	—	2	16	no	—
317	C₂×C₈	—	2	2	no	—
318	C₂×C₈	—	2	2	no	—
319	C₂×C₈	—	2	4	no	—
320	C₂×C₈	—	2	2	no	—
321	C₂×C₈	—	2	2	no	—
322	C₂×C₈	—	2	16	no	—
323	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
324	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
325	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
326	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
327	C₂×C₈	C₂×D₄×C₆₇	1	8	yes	—
328	C₄⋊C₄	—	2	8	no	—
329	C₂²⋊C₄	—	2	8	no	—
330	C₂²⋊C₄	—	2	8	no	—
331	C₂²⋊C₄	—	4	4	no	—
332	C₄²	—	2	4	no	—
333	C₄²	—	2	4	no	—
334	C₂³	—	2	2	no	—
335	C₂³	—	2	2	no	—
336	C₂³	—	2	2	no	—
337	C₂³	—	2	2	no	—
338	C₂³	—	2	2	no	—
339	C₂³	—	2	2	no	—
340	C₂³	—	2	2	no	—
341	C₂³	—	2	8	no	—
342	C₂³	—	2	8	no	—
343	C₂³	—	4	4	no	—
344	C₂³	—	4	16	no	—
345	C₂³	C₂×C₄²×C₆₇	1	2	yes	—
346	C₂³	C₂×C₄²×C₆₇	1	2	yes	—
347	C₂³	C₂×C₄²×C₆₇	1	2	yes	—
348	C₂³	C₄×D₄×C₆₇	1	4	yes	—
349	C₂³	#131×C₆₇	1	4	yes	—
350	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
351	C₂×C₄	—	2	2	no	—
352	C₂×C₄	—	2	2	no	—
353	C₂×C₄	—	2	8	no	—
354	C₂×C₄	—	2	2	no	—
355	C₂×C₄	—	2	2	no	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	2	2	no	—
358	C₂×C₄	—	2	4	no	—
359	C₂×C₄	—	2	4	no	—
360	C₂×C₄	—	2	8	no	—
361	C₂×C₄	—	2	8	no	—
362	C₂×C₄	—	2	8	no	—
363	C₂×C₄	—	2	8	no	—
364	C₂×C₄	—	2	8	no	—
365	C₂×C₄	C₂×C₄²×C₆₇	1	2	yes	—
366	C₂×C₄	C₂×C₄²×C₆₇	1	2	yes	—
367	C₂×C₄	C₂×C₄²×C₆₇	1	2	yes	—
368	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
369	C₂×C₄	#117×C₆₇	1	4	yes	—
370	C₂×C₄	C₄×D₄×C₆₇	1	8	yes	—
371	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
372	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
373	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
374	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
375	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
376	C₈	—	2	4	no	—
377	C₈	—	2	4	no	—
378	C₈	—	2	4	no	—
379	C₈	C₂×C₂²⋊C₄×C₆₇	1	8	yes	—
380	C₂²	—	2	2	no	—
381	C₂²	—	2	2	no	—
382	C₂²	—	2	2	no	—
383	C₂²	—	2	2	no	—
384	C₂²	—	2	8	no	—
385	C₂²	—	2	8	no	—
386	C₂²	—	2	2	no	—
387	C₂²	—	2	2	no	—
388	C₂²	—	2	2	no	—
389	C₂²	—	4	16	no	—
390	C₂²	—	4	4	no	—
391	C₂²	C₄×C₂²⋊C₄×C₆₇	1	4	yes	—
392	C₂²	#405×C₆₇	1	4	yes	complemented
393	C₂²	C₂²×C₄²×C₆₇	1	1	yes	characteristic
394	C₂²	C₂×C₄×D₄×C₆₇	1	2	yes	—
395	C₂²	C₂×#131×C₆₇	1	2	yes	—
396	C₂²	#568×C₆₇	1	2	yes	—
397	C₄	—	2	2	no	—
398	C₄	—	2	2	no	—
399	C₄	—	4	8	no	—
400	C₄	C₄×C₂²⋊C₄×C₆₇	1	4	yes	—
401	C₄	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
402	C₄	C₂×#117×C₆₇	1	1	yes	characteristic
403	C₄	C₂×C₄×D₄×C₆₇	1	2	yes	—
404	C₂	—	2	4	no	—
405	C₂	—	2	4	no	—
406	C₂	—	4	2	no	—
407	C₂	C₂×C₄×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
408	C₂	C₂×#405×C₆₇	1	1	yes	characteristic
409	C₂	C₆₇×#2873	1	1	yes	characteristic
410	C₂	C₆₇×#2933	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂⁴
commutator	C₂²	C₂²×C₄²×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂³×C₆₇	C₂³×C₄
prime radical	C₂⁵×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. C₂×#405×C₆₇	C₂	semidirect
5. C₆₇×#1753	C₂	semidirect
6. C₆₇×#1752	C₂	semidirect
7. C₈×C₂²⋊C₄×C₆₇	C₂	semidirect
8. C₆₇×#1729	C₂	semidirect
11. C₂²×M₁₆×C₆₇	C₄	semidirect
18. C₂×C₂²⋊C₈×C₆₇	C₄	semidirect
23. C₂×#97×C₆₇	C₂²	semidirect
24. C₂×C₄×C₈×C₆₇	C₂²	semidirect
31. #336×C₆₇	C₂²	semidirect
32. #336×C₆₇	C₂²	semidirect
109. M₁₆×C₆₇	C₂²⋊C₄	semidirect
186. C₂²×C₆₇	#405	semidirect
198. #20389	C₆₇	direct (N × H)
208. C₂×#405	C₂×C₆₇	semidirect
210. #1753	C₂×C₆₇	semidirect
211. #1752	C₂×C₆₇	semidirect
212. C₈×C₂²⋊C₄	C₂×C₆₇	semidirect
213. #1729	C₂×C₆₇	semidirect
214. C₆₇	#20389	direct (N × H)
217. C₂²×M₁₆	C₄×C₆₇	semidirect
224. C₂×C₂²⋊C₈	C₄×C₆₇	semidirect
229. C₂×#97	C₂²×C₆₇	semidirect
230. C₂×C₄×C₈	C₂²×C₆₇	semidirect
237. #336	C₂²×C₆₇	semidirect
238. #336	C₂²×C₆₇	semidirect
315. M₁₆	C₂²⋊C₄×C₆₇	semidirect
392. C₂²	#405×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂³×M₁₆×C₆₇	2
C₂²×C₂²⋊C₈×C₆₇	2
C₂×#405×C₆₇	2
C₆₇×#2280	2
C₆₇×#1753	2
C₆₇×#1752	2
C₈×C₂²⋊C₄×C₆₇	2
C₆₇×#1729	2
#20389	67

Subgroups — C67×#20389 #30699640

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#20389 #30699640