Subgroups — C₆₇×#21941 #30701192

Order 17152.

1194 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂²⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2280	C₂	1	1	yes	maximal, characteristic
3	C₂×#380×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1854	C₂	1	4	yes	maximal, complemented
5	C₆₇×#1853	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1852	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1808	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1807	C₂	1	2	yes	maximal, complemented
9	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
10	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
11	C₂×#117×C₆₇	C₂²	1	2	yes	—
12	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
13	C₂²×C₂²⋊C₄×C₆₇	—	2	1	no	—
14	C₂×#107×C₆₇	C₂²	1	4	yes	complemented
15	C₂×#106×C₆₇	C₂²	1	4	yes	complemented
16	C₂×C₂²⋊C₈×C₆₇	C₄	1	8	yes	complemented
17	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
20	#381×C₆₇	—	2	1	no	—
21	#380×C₆₇	—	2	1	no	—
22	#380×C₆₇	—	2	1	no	—
23	#380×C₆₇	C₂²	1	2	yes	—
24	#380×C₆₇	C₂²	1	2	yes	—
25	#380×C₆₇	C₂²	1	1	yes	characteristic
26	#380×C₆₇	C₂²	1	1	yes	characteristic
27	#380×C₆₇	C₂²	1	2	yes	—
28	#379×C₆₇	—	2	1	no	—
29	C₄×C₂²⋊C₄×C₆₇	—	2	1	no	—
30	C₄×C₂²⋊C₄×C₆₇	—	2	1	no	—
31	C₂×#95×C₆₇	—	2	1	no	—
32	#340×C₆₇	C₂²	1	4	yes	complemented
33	#340×C₆₇	C₂²	1	4	yes	complemented
34	C₂⁵×C₆₇	D₄	1	1	yes	characteristic
35	C₂³×C₄×C₆₇	—	2	2	no	—
36	C₂³×C₄×C₆₇	—	4	4	no	—
37	C₂³×C₄×C₆₇	C₂×C₄	1	4	yes	—
38	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
39	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
40	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
41	C₂³×C₄×C₆₇	Q₈	1	4	yes	—
42	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
43	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
44	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₆₇	—	4	4	no	—
46	C₂²×C₈×C₆₇	C₂×C₄	1	8	yes	complemented
47	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
48	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
49	#117×C₆₇	—	4	4	no	—
50	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
51	C₂×C₄⋊C₄×C₆₇	—	4	4	no	—
52	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
53	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
54	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
55	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
56	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
57	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
58	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
59	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
60	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
61	C₂×C₂²⋊C₄×C₆₇	—	4	4	no	—
62	C₂×C₄²×C₆₇	—	2	2	no	—
63	#107×C₆₇	—	4	4	no	—
64	#106×C₆₇	—	4	4	no	—
65	C₂²⋊C₈×C₆₇	C₂×C₄	1	4	yes	—
66	C₂²⋊C₈×C₆₇	C₂×C₄	1	4	yes	—
67	C₂²⋊C₈×C₆₇	C₂×C₄	1	8	yes	complemented
68	#95×C₆₇	—	2	2	no	—
69	#95×C₆₇	—	2	2	no	—
70	#95×C₆₇	—	2	2	no	—
71	#95×C₆₇	—	2	1	no	—
72	#95×C₆₇	—	2	1	no	—
73	C₂⁴×C₆₇	—	2	4	no	—
74	C₂⁴×C₆₇	—	2	2	no	—
75	C₂⁴×C₆₇	—	2	2	no	—
76	C₂⁴×C₆₇	—	4	2	no	—
77	C₂⁴×C₆₇	C₄⋊C₄	1	4	yes	—
78	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
79	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
80	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₆₇	—	2	4	no	—
82	C₂²×C₄×C₆₇	—	2	4	no	—
83	C₂²×C₄×C₆₇	—	2	4	no	—
84	C₂²×C₄×C₆₇	—	2	2	no	—
85	C₂²×C₄×C₆₇	—	2	2	no	—
86	C₂²×C₄×C₆₇	—	2	4	no	—
87	C₂²×C₄×C₆₇	—	4	8	no	—
88	C₂²×C₄×C₆₇	—	4	8	no	—
89	C₂²×C₄×C₆₇	—	4	2	no	—
90	C₂²×C₄×C₆₇	—	4	2	no	—
91	C₂²×C₄×C₆₇	—	4	2	no	—
92	C₂²×C₄×C₆₇	—	4	4	no	—
93	C₂²×C₄×C₆₇	—	4	4	no	—
94	C₂²×C₄×C₆₇	—	4	4	no	—
95	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
96	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
97	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	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	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
105	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
106	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
107	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
108	C₂×C₈×C₆₇	—	2	4	no	—
109	C₂×C₈×C₆₇	—	2	4	no	—
110	C₂×C₈×C₆₇	—	4	8	no	—
111	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
112	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
113	C₄⋊C₄×C₆₇	—	2	4	no	—
114	C₄⋊C₄×C₆₇	—	2	4	no	—
115	C₄⋊C₄×C₆₇	—	2	4	no	—
116	C₄⋊C₄×C₆₇	—	2	4	no	—
117	C₄⋊C₄×C₆₇	—	4	4	no	—
118	C₂²⋊C₄×C₆₇	—	4	4	no	—
119	C₂²⋊C₄×C₆₇	—	4	4	no	—
120	C₂²⋊C₄×C₆₇	—	4	4	no	—
121	C₂²⋊C₄×C₆₇	—	4	2	no	—
122	C₂²⋊C₄×C₆₇	—	4	4	no	—
123	C₂²⋊C₄×C₆₇	—	4	2	no	—
124	C₄²×C₆₇	—	4	4	no	—
125	C₂³×C₆₇	—	2	2	no	—
126	C₂³×C₆₇	—	2	2	no	—
127	C₂³×C₆₇	—	2	2	no	—
128	C₂³×C₆₇	—	2	1	no	—
129	C₂³×C₆₇	—	2	1	no	—
130	C₂³×C₆₇	—	2	2	no	—
131	C₂³×C₆₇	—	2	2	no	—
132	C₂³×C₆₇	—	2	1	no	—
133	C₂³×C₆₇	—	2	1	no	—
134	C₂³×C₆₇	—	2	8	no	—
135	C₂³×C₆₇	—	2	8	no	—
136	C₂³×C₆₇	—	4	4	no	—
137	C₂³×C₆₇	—	4	8	no	—
138	C₂³×C₆₇	—	4	8	no	—
139	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
140	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
141	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
142	C₂³×C₆₇	#135	1	4	yes	—
143	C₂³×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
144	C₂³×C₆₇	#137	1	2	yes	—
145	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄×C₆₇	—	2	2	no	—
147	C₂×C₄×C₆₇	—	2	2	no	—
148	C₂×C₄×C₆₇	—	2	2	no	—
149	C₂×C₄×C₆₇	—	2	4	no	—
150	C₂×C₄×C₆₇	—	2	4	no	—
151	C₂×C₄×C₆₇	—	2	4	no	—
152	C₂×C₄×C₆₇	—	2	4	no	—
153	C₂×C₄×C₆₇	—	2	4	no	—
154	C₂×C₄×C₆₇	—	2	4	no	—
155	C₂×C₄×C₆₇	—	4	4	no	—
156	C₂×C₄×C₆₇	—	4	4	no	—
157	C₂×C₄×C₆₇	—	4	4	no	—
158	C₂×C₄×C₆₇	—	4	2	no	—
159	C₂×C₄×C₆₇	—	4	2	no	—
160	C₂×C₄×C₆₇	—	4	4	no	—
161	C₂×C₄×C₆₇	—	4	2	no	—
162	C₂×C₄×C₆₇	—	4	2	no	—
163	C₂×C₄×C₆₇	—	4	4	no	—
164	C₂×C₄×C₆₇	—	4	4	no	—
165	C₂×C₄×C₆₇	—	4	2	no	—
166	C₂×C₄×C₆₇	—	4	2	no	—
167	C₂×C₄×C₆₇	—	8	8	no	—
168	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
169	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
170	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
171	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
172	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
173	C₂×C₄×C₆₇	H(5,𝔽₂)	1	4	yes	—
174	C₈×C₆₇	—	2	4	no	—
175	C₈×C₆₇	—	2	4	no	—
176	C₂²×C₆₇	—	2	2	no	—
177	C₂²×C₆₇	—	2	2	no	—
178	C₂²×C₆₇	—	2	2	no	—
179	C₂²×C₆₇	—	2	2	no	—
180	C₂²×C₆₇	—	2	4	no	—
181	C₂²×C₆₇	—	2	4	no	—
182	C₂²×C₆₇	—	2	4	no	—
183	C₂²×C₆₇	—	2	4	no	—
184	C₂²×C₆₇	—	2	2	no	—
185	C₂²×C₆₇	—	2	2	no	—
186	C₂²×C₆₇	—	2	2	no	—
187	C₂²×C₆₇	—	4	16	no	—
188	C₂²×C₆₇	—	4	2	no	—
189	C₂²×C₆₇	—	4	2	no	—
190	C₂²×C₆₇	#427	1	4	yes	—
191	C₂²×C₆₇	#428	1	4	yes	—
192	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
193	C₂²×C₆₇	#534	1	2	yes	—
194	C₂²×C₆₇	C₂×#135	1	2	yes	—
195	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
196	C₂²×C₆₇	C₂×#137	1	1	yes	characteristic
197	C₄×C₆₇	—	2	2	no	—
198	C₄×C₆₇	—	2	2	no	—
199	C₄×C₆₇	—	2	2	no	—
200	C₄×C₆₇	—	8	2	no	—
201	C₄×C₆₇	—	8	4	no	—
202	C₄×C₆₇	—	8	2	no	—
203	C₄×C₆₇	#518	1	2	yes	—
204	C₄×C₆₇	#543	1	2	yes	—
205	#21941	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
206	C₂×C₆₇	—	2	4	no	—
207	C₂×C₆₇	—	2	4	no	—
208	C₂×C₆₇	—	4	2	no	—
209	C₂×C₆₇	#2315	1	1	yes	characteristic
210	C₂×C₆₇	C₂×#427	1	1	yes	characteristic
211	C₂×C₆₇	C₂×#428	1	1	yes	characteristic
212	C₂×C₆₇	#3188	1	2	yes	—
213	C₂×C₆₇	#3190	1	2	yes	—
214	C₂²×C₂²⋊C₈	C₂×C₆₇	1	1	yes	characteristic
215	#2280	C₂×C₆₇	1	1	yes	characteristic
216	C₂×#380	C₂×C₆₇	1	1	yes	characteristic
217	#1854	C₂×C₆₇	1	4	yes	complemented
218	#1853	C₂×C₆₇	1	2	yes	complemented
219	#1852	C₂×C₆₇	1	2	yes	complemented
220	#1808	C₂×C₆₇	1	2	yes	complemented
221	#1807	C₂×C₆₇	1	2	yes	complemented
222	C₆₇	#21941	1	1	yes	characteristic, direct product factor, Sylow
223	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
224	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
225	C₂×#117	C₂²×C₆₇	1	2	yes	—
226	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
227	C₂²×C₂²⋊C₄	—	2	1	no	—
228	C₂×#107	C₂²×C₆₇	1	4	yes	complemented
229	C₂×#106	C₂²×C₆₇	1	4	yes	complemented
230	C₂×C₂²⋊C₈	C₄×C₆₇	1	8	yes	complemented
231	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
232	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
233	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
234	#381	—	2	1	no	—
235	#380	—	2	1	no	—
236	#380	—	2	1	no	—
237	#380	C₂²×C₆₇	1	1	yes	characteristic
238	#380	C₂²×C₆₇	1	2	yes	—
239	#380	C₂²×C₆₇	1	1	yes	characteristic
240	#380	C₂²×C₆₇	1	2	yes	—
241	#380	C₂²×C₆₇	1	2	yes	—
242	#379	—	2	1	no	—
243	C₄×C₂²⋊C₄	—	2	1	no	—
244	C₄×C₂²⋊C₄	—	2	1	no	—
245	C₂×#95	—	2	1	no	—
246	#340	C₂²×C₆₇	1	4	yes	complemented
247	#340	C₂²×C₆₇	1	4	yes	complemented
248	C₂⁵	D₄×C₆₇	1	1	yes	characteristic
249	C₂³×C₄	—	2	2	no	—
250	C₂³×C₄	—	4	4	no	—
251	C₂³×C₄	C₂×C₄×C₆₇	1	4	yes	—
252	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
253	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
254	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
255	C₂³×C₄	Q₈×C₆₇	1	4	yes	—
256	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
257	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
258	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
259	C₂²×C₈	—	4	4	no	—
260	C₂²×C₈	C₂×C₄×C₆₇	1	8	yes	complemented
261	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
262	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
263	#117	—	4	4	no	—
264	C₂×C₄⋊C₄	—	2	2	no	—
265	C₂×C₄⋊C₄	—	4	4	no	—
266	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
267	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
268	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
269	C₂×C₂²⋊C₄	—	2	2	no	—
270	C₂×C₂²⋊C₄	—	2	2	no	—
271	C₂×C₂²⋊C₄	—	2	2	no	—
272	C₂×C₂²⋊C₄	—	2	2	no	—
273	C₂×C₂²⋊C₄	—	2	1	no	—
274	C₂×C₂²⋊C₄	—	2	1	no	—
275	C₂×C₂²⋊C₄	—	4	4	no	—
276	C₂×C₄²	—	2	2	no	—
277	#107	—	4	4	no	—
278	#106	—	4	4	no	—
279	C₂²⋊C₈	C₂×C₄×C₆₇	1	4	yes	—
280	C₂²⋊C₈	C₂×C₄×C₆₇	1	4	yes	—
281	C₂²⋊C₈	C₂×C₄×C₆₇	1	8	yes	complemented
282	#95	—	2	2	no	—
283	#95	—	2	2	no	—
284	#95	—	2	1	no	—
285	#95	—	2	1	no	—
286	#95	—	2	2	no	—
287	C₂⁴	—	2	4	no	—
288	C₂⁴	—	2	2	no	—
289	C₂⁴	—	2	2	no	—
290	C₂⁴	—	4	2	no	—
291	C₂⁴	C₄⋊C₄×C₆₇	1	4	yes	—
292	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
293	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
294	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
295	C₂²×C₄	—	2	4	no	—
296	C₂²×C₄	—	2	4	no	—
297	C₂²×C₄	—	2	4	no	—
298	C₂²×C₄	—	2	2	no	—
299	C₂²×C₄	—	2	2	no	—
300	C₂²×C₄	—	2	4	no	—
301	C₂²×C₄	—	4	8	no	—
302	C₂²×C₄	—	4	8	no	—
303	C₂²×C₄	—	4	2	no	—
304	C₂²×C₄	—	4	2	no	—
305	C₂²×C₄	—	4	2	no	—
306	C₂²×C₄	—	4	4	no	—
307	C₂²×C₄	—	4	4	no	—
308	C₂²×C₄	—	4	4	no	—
309	C₂²×C₄	C₄⋊C₄×C₆₇	1	4	yes	—
310	C₂²×C₄	C₄⋊C₄×C₆₇	1	4	yes	—
311	C₂²×C₄	C₄⋊C₄×C₆₇	1	4	yes	—
312	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
313	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
314	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
315	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
316	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
317	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
318	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
319	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
320	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
321	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
322	C₂×C₈	—	2	4	no	—
323	C₂×C₈	—	2	4	no	—
324	C₂×C₈	—	4	8	no	—
325	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
326	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
327	C₄⋊C₄	—	2	4	no	—
328	C₄⋊C₄	—	2	4	no	—
329	C₄⋊C₄	—	2	4	no	—
330	C₄⋊C₄	—	2	4	no	—
331	C₄⋊C₄	—	4	4	no	—
332	C₂²⋊C₄	—	4	4	no	—
333	C₂²⋊C₄	—	4	4	no	—
334	C₂²⋊C₄	—	4	4	no	—
335	C₂²⋊C₄	—	4	4	no	—
336	C₂²⋊C₄	—	4	2	no	—
337	C₂²⋊C₄	—	4	2	no	—
338	C₄²	—	4	4	no	—
339	C₂³	—	2	2	no	—
340	C₂³	—	2	2	no	—
341	C₂³	—	2	2	no	—
342	C₂³	—	2	1	no	—
343	C₂³	—	2	1	no	—
344	C₂³	—	2	2	no	—
345	C₂³	—	2	2	no	—
346	C₂³	—	2	1	no	—
347	C₂³	—	2	1	no	—
348	C₂³	—	2	8	no	—
349	C₂³	—	2	8	no	—
350	C₂³	—	4	4	no	—
351	C₂³	—	4	8	no	—
352	C₂³	—	4	8	no	—
353	C₂³	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
354	C₂³	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
355	C₂³	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
356	C₂³	#135×C₆₇	1	4	yes	—
357	C₂³	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
358	C₂³	#137×C₆₇	1	2	yes	—
359	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
360	C₂×C₄	—	2	2	no	—
361	C₂×C₄	—	2	2	no	—
362	C₂×C₄	—	2	2	no	—
363	C₂×C₄	—	2	4	no	—
364	C₂×C₄	—	2	4	no	—
365	C₂×C₄	—	2	4	no	—
366	C₂×C₄	—	2	4	no	—
367	C₂×C₄	—	2	4	no	—
368	C₂×C₄	—	2	4	no	—
369	C₂×C₄	—	4	4	no	—
370	C₂×C₄	—	4	4	no	—
371	C₂×C₄	—	4	4	no	—
372	C₂×C₄	—	4	2	no	—
373	C₂×C₄	—	4	2	no	—
374	C₂×C₄	—	4	4	no	—
375	C₂×C₄	—	4	2	no	—
376	C₂×C₄	—	4	2	no	—
377	C₂×C₄	—	4	4	no	—
378	C₂×C₄	—	4	4	no	—
379	C₂×C₄	—	4	2	no	—
380	C₂×C₄	—	4	2	no	—
381	C₂×C₄	—	8	8	no	—
382	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
383	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
384	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
385	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
386	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
387	C₂×C₄	H(5,𝔽₂)×C₆₇	1	4	yes	—
388	C₈	—	2	4	no	—
389	C₈	—	2	4	no	—
390	C₂²	—	2	2	no	—
391	C₂²	—	2	2	no	—
392	C₂²	—	2	2	no	—
393	C₂²	—	2	2	no	—
394	C₂²	—	2	4	no	—
395	C₂²	—	2	4	no	—
396	C₂²	—	2	4	no	—
397	C₂²	—	2	4	no	—
398	C₂²	—	2	2	no	—
399	C₂²	—	2	2	no	—
400	C₂²	—	2	2	no	—
401	C₂²	—	4	16	no	—
402	C₂²	—	4	2	no	—
403	C₂²	—	4	2	no	—
404	C₂²	#427×C₆₇	1	4	yes	—
405	C₂²	#428×C₆₇	1	4	yes	—
406	C₂²	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
407	C₂²	#534×C₆₇	1	2	yes	—
408	C₂²	C₂×#135×C₆₇	1	2	yes	—
409	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
410	C₂²	C₂×#137×C₆₇	1	1	yes	characteristic
411	C₄	—	2	2	no	—
412	C₄	—	2	2	no	—
413	C₄	—	2	2	no	—
414	C₄	—	8	4	no	—
415	C₄	—	8	2	no	—
416	C₄	—	8	2	no	—
417	C₄	#518×C₆₇	1	2	yes	—
418	C₄	#543×C₆₇	1	2	yes	—
419	C₂	—	2	4	no	—
420	C₂	—	2	4	no	—
421	C₂	—	4	2	no	—
422	C₂	C₆₇×#2315	1	1	yes	characteristic
423	C₂	C₂×#427×C₆₇	1	1	yes	characteristic
424	C₂	C₂×#428×C₆₇	1	1	yes	characteristic
425	C₂	C₆₇×#3188	1	2	yes	—
426	C₂	C₆₇×#3190	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂⁵×C₆₇	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂⁴×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₆₇×#1854	C₂	semidirect
5. C₆₇×#1853	C₂	semidirect
6. C₆₇×#1852	C₂	semidirect
7. C₆₇×#1808	C₂	semidirect
8. C₆₇×#1807	C₂	semidirect
14. C₂×#107×C₆₇	C₂²	semidirect
15. C₂×#106×C₆₇	C₂²	semidirect
16. C₂×C₂²⋊C₈×C₆₇	C₄	semidirect
32. #340×C₆₇	C₂²	semidirect
33. #340×C₆₇	C₂²	semidirect
46. C₂²×C₈×C₆₇	C₂×C₄	semidirect
67. C₂²⋊C₈×C₆₇	C₂×C₄	semidirect
205. #21941	C₆₇	direct (N × H)
217. #1854	C₂×C₆₇	semidirect
218. #1853	C₂×C₆₇	semidirect
219. #1852	C₂×C₆₇	semidirect
220. #1808	C₂×C₆₇	semidirect
221. #1807	C₂×C₆₇	semidirect
222. C₆₇	#21941	direct (N × H)
228. C₂×#107	C₂²×C₆₇	semidirect
229. C₂×#106	C₂²×C₆₇	semidirect
230. C₂×C₂²⋊C₈	C₄×C₆₇	semidirect
246. #340	C₂²×C₆₇	semidirect
247. #340	C₂²×C₆₇	semidirect
260. C₂²×C₈	C₂×C₄×C₆₇	semidirect
281. C₂²⋊C₈	C₂×C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₂²⋊C₈×C₆₇	2
C₆₇×#2280	2
C₂×#380×C₆₇	2
C₆₇×#1854	2
C₆₇×#1853	2
C₆₇×#1852	2
C₆₇×#1808	2
C₆₇×#1807	2
#21941	67

Subgroups — C67×#21941 #30701192

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21941 #30701192