Subgroups — C₆₇×#10849 #30690100

Order 17152.

1086 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#384×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#340×C₆₇	C₂	1	4	yes	maximal
5	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
6	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
7	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
8	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
9	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
10	C₂²×C₄²×C₆₇	—	2	1	no	—
11	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₈×C₆₇	—	2	2	no	—
13	C₂×C₄⋊C₈×C₆₇	—	2	2	no	—
14	C₂×C₄⋊C₈×C₆₇	C₄	1	4	yes	complemented
15	#384×C₆₇	—	2	2	no	—
16	#384×C₆₇	C₄	1	4	yes	complemented
17	C₄×C₄⋊C₄×C₆₇	—	2	2	no	—
18	C₄×C₄⋊C₄×C₆₇	C₄	1	4	yes	complemented
19	C₂×#95×C₆₇	—	2	1	no	—
20	#340×C₆₇	—	2	8	no	—
21	C₂³×C₄×C₆₇	—	2	2	no	—
22	C₂³×C₄×C₆₇	—	2	2	no	—
23	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
24	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
26	C₂²×C₈×C₆₇	—	2	4	no	—
27	C₂²×C₈×C₆₇	—	2	2	no	—
28	C₂²×C₈×C₆₇	—	2	2	no	—
29	C₂²×C₈×C₆₇	—	2	4	no	—
30	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
31	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
32	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
33	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
34	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
35	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
36	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
37	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
38	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
39	C₂×C₄⋊C₄×C₆₇	—	4	4	no	—
40	C₂×C₄⋊C₄×C₆₇	—	4	2	no	—
41	C₂×C₄⋊C₄×C₆₇	—	4	2	no	—
42	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
43	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
44	C₂×C₄⋊C₄×C₆₇	D₄	1	4	yes	—
45	C₂×C₄⋊C₄×C₆₇	Q₈	1	4	yes	—
46	C₂×C₄²×C₆₇	—	2	4	no	—
47	C₂×C₄²×C₆₇	—	4	2	no	—
48	C₂×C₄²×C₆₇	—	4	2	no	—
49	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
51	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₆₇	D₄	1	2	yes	—
53	C₂×C₄²×C₆₇	D₄	1	2	yes	—
54	C₂×C₄²×C₆₇	Q₈	1	2	yes	—
55	C₄⋊C₈×C₆₇	—	2	8	no	—
56	#95×C₆₇	—	2	4	no	—
57	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₆₇	—	2	2	no	—
59	C₂²×C₄×C₆₇	—	2	2	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	—	4	2	no	—
65	C₂²×C₄×C₆₇	—	4	2	no	—
66	C₂²×C₄×C₆₇	—	4	2	no	—
67	C₂²×C₄×C₆₇	—	4	2	no	—
68	C₂²×C₄×C₆₇	—	4	4	no	—
69	C₂²×C₄×C₆₇	—	4	2	no	—
70	C₂²×C₄×C₆₇	—	4	2	no	—
71	C₂²×C₄×C₆₇	—	4	4	no	—
72	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
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	C₄⋊C₄	1	2	yes	—
77	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
78	C₂²×C₄×C₆₇	D₈	1	2	yes	—
79	C₂²×C₄×C₆₇	SD₁₆	1	2	yes	—
80	C₂²×C₄×C₆₇	SD₁₆	1	2	yes	—
81	C₂²×C₄×C₆₇	Q₁₆	1	2	yes	—
82	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
88	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
89	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
90	C₂²×C₄×C₆₇	#41	1	2	yes	—
91	C₂²×C₄×C₆₇	#41	1	2	yes	—
92	C₂×C₈×C₆₇	—	2	4	no	—
93	C₂×C₈×C₆₇	—	2	8	no	—
94	C₂×C₈×C₆₇	—	4	4	no	—
95	C₂×C₈×C₆₇	—	4	2	no	—
96	C₂×C₈×C₆₇	—	4	2	no	—
97	C₄⋊C₄×C₆₇	—	2	4	no	—
98	C₄⋊C₄×C₆₇	—	2	4	no	—
99	C₄⋊C₄×C₆₇	—	2	4	no	—
100	C₄⋊C₄×C₆₇	—	2	8	no	—
101	C₄⋊C₄×C₆₇	—	4	8	no	—
102	C₄⋊C₄×C₆₇	—	4	8	no	—
103	C₄⋊C₄×C₆₇	—	4	2	no	—
104	C₄⋊C₄×C₆₇	—	4	2	no	—
105	C₄⋊C₄×C₆₇	C₄⋊C₄	1	8	yes	complemented
106	C₄²×C₆₇	—	4	8	no	—
107	C₄²×C₆₇	C₄²	1	4	yes	—
108	C₄²×C₆₇	C₂²⋊C₄	1	4	yes	—
109	C₄²×C₆₇	C₄⋊C₄	1	4	yes	—
110	C₄²×C₆₇	C₄⋊C₄	1	4	yes	—
111	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂³×C₆₇	#121	1	2	yes	—
115	C₂³×C₆₇	C₂²⋊Q₈	1	2	yes	—
116	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
117	C₂³×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
118	C₂³×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
119	C₂³×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
120	C₂³×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
121	C₂³×C₆₇	#137	1	2	yes	—
122	C₂×C₄×C₆₇	—	2	4	no	—
123	C₂×C₄×C₆₇	—	2	2	no	—
124	C₂×C₄×C₆₇	—	2	2	no	—
125	C₂×C₄×C₆₇	—	2	4	no	—
126	C₂×C₄×C₆₇	—	2	8	no	—
127	C₂×C₄×C₆₇	—	4	4	no	—
128	C₂×C₄×C₆₇	—	4	4	no	—
129	C₂×C₄×C₆₇	—	4	4	no	—
130	C₂×C₄×C₆₇	—	4	4	no	—
131	C₂×C₄×C₆₇	—	4	4	no	—
132	C₂×C₄×C₆₇	—	4	4	no	—
133	C₂×C₄×C₆₇	—	4	4	no	—
134	C₂×C₄×C₆₇	—	4	4	no	—
135	C₂×C₄×C₆₇	—	4	4	no	—
136	C₂×C₄×C₆₇	—	8	4	no	—
137	C₂×C₄×C₆₇	—	8	4	no	—
138	C₂×C₄×C₆₇	#95	1	4	yes	—
139	C₂×C₄×C₆₇	#95	1	4	yes	—
140	C₂×C₄×C₆₇	#102	1	4	yes	—
141	C₂×C₄×C₆₇	#102	1	4	yes	—
142	C₂×C₄×C₆₇	#103	1	4	yes	—
143	C₂×C₄×C₆₇	#103	1	4	yes	—
144	C₂×C₄×C₆₇	#106	1	4	yes	—
145	C₂×C₄×C₆₇	#107	1	4	yes	—
146	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
147	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
148	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
149	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
150	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
151	C₂×C₄×C₆₇	#117	1	2	yes	—
152	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
153	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
154	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
155	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
156	C₂×C₄×C₆₇	#121	1	2	yes	—
157	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
158	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
159	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
160	C₂×C₄×C₆₇	#125	1	2	yes	—
161	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
162	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
163	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
164	C₈×C₆₇	—	4	4	no	—
165	C₂²×C₆₇	C₂×#95	1	1	yes	characteristic
166	C₂²×C₆₇	#380	1	2	yes	—
167	C₂²×C₆₇	#389	1	2	yes	—
168	C₂²×C₆₇	#390	1	1	yes	characteristic
169	C₂²×C₆₇	#391	1	1	yes	characteristic
170	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
171	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
172	C₂²×C₆₇	C₂×#103	1	1	yes	characteristic
173	C₂²×C₆₇	C₂×#103	1	1	yes	characteristic
174	C₂²×C₆₇	#417	1	2	yes	—
175	C₂²×C₆₇	#418	1	1	yes	characteristic
176	C₂²×C₆₇	#419	1	1	yes	characteristic
177	C₂²×C₆₇	C₂×#106	1	1	yes	characteristic
178	C₂²×C₆₇	C₂×#107	1	1	yes	characteristic
179	C₂²×C₆₇	#428	1	2	yes	—
180	C₂²×C₆₇	#459	1	2	yes	—
181	C₂²×C₆₇	#460	1	2	yes	—
182	C₂²×C₆₇	#461	1	2	yes	—
183	C₂²×C₆₇	#462	1	2	yes	—
184	C₂²×C₆₇	#474	1	2	yes	—
185	C₂²×C₆₇	#475	1	2	yes	—
186	C₂²×C₆₇	#476	1	2	yes	—
187	C₂²×C₆₇	#477	1	2	yes	—
188	C₄×C₆₇	—	2	4	no	—
189	C₄×C₆₇	—	4	8	no	—
190	C₄×C₆₇	—	8	4	no	—
191	C₄×C₆₇	#340	1	8	yes	complemented
192	C₄×C₆₇	C₄×C₄⋊C₄	1	2	yes	—
193	C₄×C₆₇	#382	1	1	yes	characteristic
194	C₄×C₆₇	#384	1	1	yes	characteristic
195	C₄×C₆₇	#389	1	2	yes	—
196	C₄×C₆₇	#390	1	1	yes	characteristic
197	C₄×C₆₇	#391	1	1	yes	characteristic
198	#10849	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
199	C₂×C₆₇	C₄⋊#95	1	1	yes	characteristic
200	C₂×C₆₇	C₂×#340	1	1	yes	characteristic
201	C₂×C₆₇	#1736	1	1	yes	characteristic
202	C₂×C₆₇	#1802	1	2	yes	—
203	C₂×C₆₇	#1803	1	2	yes	—
204	C₂×C₆₇	#1947	1	2	yes	—
205	C₂×C₆₇	#1948	1	2	yes	—
206	C₂×C₆₇	#1981	1	1	yes	characteristic
207	C₂×C₆₇	#1982	1	1	yes	characteristic
208	C₂×C₆₇	#1987	1	1	yes	characteristic
209	C₂×C₆₇	#1988	1	1	yes	characteristic
210	C₂²×C₄⋊C₈	C₂×C₆₇	1	1	yes	characteristic
211	C₂×#384	C₂×C₆₇	1	1	yes	characteristic
212	C₂×C₄×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic
213	C₂×#340	C₂×C₆₇	1	4	yes	—
214	C₆₇	#10849	1	1	yes	characteristic, direct product factor, Sylow
215	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
216	C₂²×C₄⋊C₄	—	2	1	no	—
217	C₂²×C₄⋊C₄	—	2	1	no	—
218	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
219	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
220	C₂²×C₄²	—	2	1	no	—
221	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
222	C₂×C₄⋊C₈	—	2	2	no	—
223	C₂×C₄⋊C₈	—	2	2	no	—
224	C₂×C₄⋊C₈	C₄×C₆₇	1	4	yes	complemented
225	#384	—	2	2	no	—
226	#384	C₄×C₆₇	1	4	yes	complemented
227	C₄×C₄⋊C₄	—	2	2	no	—
228	C₄×C₄⋊C₄	C₄×C₆₇	1	4	yes	complemented
229	C₂×#95	—	2	1	no	—
230	#340	—	2	8	no	—
231	C₂³×C₄	—	2	2	no	—
232	C₂³×C₄	—	2	2	no	—
233	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
234	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
235	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
236	C₂²×C₈	—	2	4	no	—
237	C₂²×C₈	—	2	2	no	—
238	C₂²×C₈	—	2	2	no	—
239	C₂²×C₈	—	2	4	no	—
240	C₂²×C₈	C₂×C₄×C₆₇	1	4	yes	—
241	C₂×C₄⋊C₄	—	2	2	no	—
242	C₂×C₄⋊C₄	—	2	2	no	—
243	C₂×C₄⋊C₄	—	2	2	no	—
244	C₂×C₄⋊C₄	—	2	4	no	—
245	C₂×C₄⋊C₄	—	2	4	no	—
246	C₂×C₄⋊C₄	—	2	4	no	—
247	C₂×C₄⋊C₄	—	2	4	no	—
248	C₂×C₄⋊C₄	—	2	2	no	—
249	C₂×C₄⋊C₄	—	4	4	no	—
250	C₂×C₄⋊C₄	—	4	2	no	—
251	C₂×C₄⋊C₄	—	4	2	no	—
252	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
253	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
254	C₂×C₄⋊C₄	D₄×C₆₇	1	4	yes	—
255	C₂×C₄⋊C₄	Q₈×C₆₇	1	4	yes	—
256	C₂×C₄²	—	2	4	no	—
257	C₂×C₄²	—	4	2	no	—
258	C₂×C₄²	—	4	2	no	—
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	2	yes	—
262	C₂×C₄²	D₄×C₆₇	1	2	yes	—
263	C₂×C₄²	D₄×C₆₇	1	2	yes	—
264	C₂×C₄²	Q₈×C₆₇	1	2	yes	—
265	C₄⋊C₈	—	2	8	no	—
266	#95	—	2	4	no	—
267	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
268	C₂²×C₄	—	2	2	no	—
269	C₂²×C₄	—	2	2	no	—
270	C₂²×C₄	—	2	4	no	—
271	C₂²×C₄	—	2	4	no	—
272	C₂²×C₄	—	2	4	no	—
273	C₂²×C₄	—	2	4	no	—
274	C₂²×C₄	—	4	2	no	—
275	C₂²×C₄	—	4	2	no	—
276	C₂²×C₄	—	4	2	no	—
277	C₂²×C₄	—	4	2	no	—
278	C₂²×C₄	—	4	4	no	—
279	C₂²×C₄	—	4	2	no	—
280	C₂²×C₄	—	4	2	no	—
281	C₂²×C₄	—	4	4	no	—
282	C₂²×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
283	C₂²×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
284	C₂²×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
285	C₂²×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
286	C₂²×C₄	C₄⋊C₄×C₆₇	1	2	yes	—
287	C₂²×C₄	C₄⋊C₄×C₆₇	1	2	yes	—
288	C₂²×C₄	D₈×C₆₇	1	2	yes	—
289	C₂²×C₄	SD₁₆×C₆₇	1	2	yes	—
290	C₂²×C₄	SD₁₆×C₆₇	1	2	yes	—
291	C₂²×C₄	Q₁₆×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	1	yes	characteristic
296	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
297	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
298	C₂²×C₄	C₂×Q₈×C₆₇	1	1	yes	characteristic
299	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
300	C₂²×C₄	#41×C₆₇	1	2	yes	—
301	C₂²×C₄	#41×C₆₇	1	2	yes	—
302	C₂×C₈	—	2	4	no	—
303	C₂×C₈	—	2	8	no	—
304	C₂×C₈	—	4	4	no	—
305	C₂×C₈	—	4	2	no	—
306	C₂×C₈	—	4	2	no	—
307	C₄⋊C₄	—	2	4	no	—
308	C₄⋊C₄	—	2	4	no	—
309	C₄⋊C₄	—	2	4	no	—
310	C₄⋊C₄	—	2	8	no	—
311	C₄⋊C₄	—	4	8	no	—
312	C₄⋊C₄	—	4	8	no	—
313	C₄⋊C₄	—	4	2	no	—
314	C₄⋊C₄	—	4	2	no	—
315	C₄⋊C₄	C₄⋊C₄×C₆₇	1	8	yes	complemented
316	C₄²	—	4	8	no	—
317	C₄²	C₄²×C₆₇	1	4	yes	—
318	C₄²	C₂²⋊C₄×C₆₇	1	4	yes	—
319	C₄²	C₄⋊C₄×C₆₇	1	4	yes	—
320	C₄²	C₄⋊C₄×C₆₇	1	4	yes	—
321	C₂³	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
322	C₂³	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
323	C₂³	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
324	C₂³	#121×C₆₇	1	2	yes	—
325	C₂³	C₂²⋊Q₈×C₆₇	1	2	yes	—
326	C₂³	C₂×D₈×C₆₇	1	1	yes	characteristic
327	C₂³	C₂×SD₁₆×C₆₇	1	1	yes	characteristic
328	C₂³	C₂×SD₁₆×C₆₇	1	1	yes	characteristic
329	C₂³	C₂×Q₁₆×C₆₇	1	1	yes	characteristic
330	C₂³	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
331	C₂³	#137×C₆₇	1	2	yes	—
332	C₂×C₄	—	2	4	no	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	2	4	no	—
336	C₂×C₄	—	2	8	no	—
337	C₂×C₄	—	4	4	no	—
338	C₂×C₄	—	4	4	no	—
339	C₂×C₄	—	4	4	no	—
340	C₂×C₄	—	4	4	no	—
341	C₂×C₄	—	4	4	no	—
342	C₂×C₄	—	4	4	no	—
343	C₂×C₄	—	4	4	no	—
344	C₂×C₄	—	4	4	no	—
345	C₂×C₄	—	4	4	no	—
346	C₂×C₄	—	8	4	no	—
347	C₂×C₄	—	8	4	no	—
348	C₂×C₄	#95×C₆₇	1	4	yes	—
349	C₂×C₄	#95×C₆₇	1	4	yes	—
350	C₂×C₄	#102×C₆₇	1	4	yes	—
351	C₂×C₄	#102×C₆₇	1	4	yes	—
352	C₂×C₄	#103×C₆₇	1	4	yes	—
353	C₂×C₄	#103×C₆₇	1	4	yes	—
354	C₂×C₄	#106×C₆₇	1	4	yes	—
355	C₂×C₄	#107×C₆₇	1	4	yes	—
356	C₂×C₄	C₂×C₄²×C₆₇	1	1	yes	characteristic
357	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
358	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
359	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
360	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
361	C₂×C₄	#117×C₆₇	1	2	yes	—
362	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
363	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
364	C₂×C₄	C₄×Q₈×C₆₇	1	2	yes	—
365	C₂×C₄	C₄×Q₈×C₆₇	1	2	yes	—
366	C₂×C₄	#121×C₆₇	1	2	yes	—
367	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
368	C₂×C₄	#124×C₆₇	1	1	yes	characteristic
369	C₂×C₄	#124×C₆₇	1	1	yes	characteristic
370	C₂×C₄	#125×C₆₇	1	2	yes	—
371	C₂×C₄	#127×C₆₇	1	1	yes	characteristic
372	C₂×C₄	C₄⋊Q₈×C₆₇	1	2	yes	—
373	C₂×C₄	C₄⋊Q₈×C₆₇	1	1	yes	characteristic
374	C₈	—	4	4	no	—
375	C₂²	C₂×#95×C₆₇	1	1	yes	characteristic
376	C₂²	#380×C₆₇	1	2	yes	—
377	C₂²	#389×C₆₇	1	2	yes	—
378	C₂²	#390×C₆₇	1	1	yes	characteristic
379	C₂²	#391×C₆₇	1	1	yes	characteristic
380	C₂²	C₂×#102×C₆₇	1	1	yes	characteristic
381	C₂²	C₂×#102×C₆₇	1	1	yes	characteristic
382	C₂²	C₂×#103×C₆₇	1	1	yes	characteristic
383	C₂²	C₂×#103×C₆₇	1	1	yes	characteristic
384	C₂²	#417×C₆₇	1	2	yes	—
385	C₂²	#418×C₆₇	1	1	yes	characteristic
386	C₂²	#419×C₆₇	1	1	yes	characteristic
387	C₂²	C₂×#106×C₆₇	1	1	yes	characteristic
388	C₂²	C₂×#107×C₆₇	1	1	yes	characteristic
389	C₂²	#428×C₆₇	1	2	yes	—
390	C₂²	#459×C₆₇	1	2	yes	—
391	C₂²	#460×C₆₇	1	2	yes	—
392	C₂²	#461×C₆₇	1	2	yes	—
393	C₂²	#462×C₆₇	1	2	yes	—
394	C₂²	#474×C₆₇	1	2	yes	—
395	C₂²	#475×C₆₇	1	2	yes	—
396	C₂²	#476×C₆₇	1	2	yes	—
397	C₂²	#477×C₆₇	1	2	yes	—
398	C₄	—	2	4	no	—
399	C₄	—	4	8	no	—
400	C₄	—	8	4	no	—
401	C₄	#340×C₆₇	1	8	yes	complemented
402	C₄	C₄×C₄⋊C₄×C₆₇	1	2	yes	—
403	C₄	#382×C₆₇	1	1	yes	characteristic
404	C₄	#384×C₆₇	1	1	yes	characteristic
405	C₄	#389×C₆₇	1	2	yes	—
406	C₄	#390×C₆₇	1	1	yes	characteristic
407	C₄	#391×C₆₇	1	1	yes	characteristic
408	C₂	C₆₇×C₄⋊#95	1	1	yes	characteristic
409	C₂	C₂×#340×C₆₇	1	1	yes	characteristic
410	C₂	C₆₇×#1736	1	1	yes	characteristic
411	C₂	C₆₇×#1802	1	2	yes	—
412	C₂	C₆₇×#1803	1	2	yes	—
413	C₂	C₆₇×#1947	1	2	yes	—
414	C₂	C₆₇×#1948	1	2	yes	—
415	C₂	C₆₇×#1981	1	1	yes	characteristic
416	C₂	C₆₇×#1982	1	1	yes	characteristic
417	C₂	C₆₇×#1987	1	1	yes	characteristic
418	C₂	C₆₇×#1988	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₆₇	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₆₇
Frattini, cosocle	C₂³×C₄	C₂³×C₆₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
14. C₂×C₄⋊C₈×C₆₇	C₄	semidirect
16. #384×C₆₇	C₄	semidirect
18. C₄×C₄⋊C₄×C₆₇	C₄	semidirect
105. C₄⋊C₄×C₆₇	C₄⋊C₄	semidirect
191. C₄×C₆₇	#340	semidirect
198. #10849	C₆₇	direct (N × H)
214. C₆₇	#10849	direct (N × H)
224. C₂×C₄⋊C₈	C₄×C₆₇	semidirect
226. #384	C₄×C₆₇	semidirect
228. C₄×C₄⋊C₄	C₄×C₆₇	semidirect
315. C₄⋊C₄	C₄⋊C₄×C₆₇	semidirect
401. C₄	#340×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊C₈×C₆₇	2
C₂×#384×C₆₇	2
C₂×C₄×C₄⋊C₄×C₆₇	2
C₂×#340×C₆₇	2
#10849	67

Subgroups — C67×#10849 #30690100

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#10849 #30690100