Subgroups — C₆₇×#21894 #30701145

Order 17152.

1130 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#533×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#423×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2285	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1988	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1981	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1894	C₂	1	4	yes	maximal
7	C₆₇×#1803	C₂	1	4	yes	maximal, complemented
8	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
9	#533×C₆₇	—	2	4	no	—
10	#533×C₆₇	C₄	1	8	yes	complemented
11	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×#125×C₆₇	—	2	1	no	—
14	C₂×C₂²⋊Q₈×C₆₇	—	2	1	no	—
15	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
16	C₂×C₄×Q₈×C₆₇	—	2	1	no	—
17	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
18	C₂×#117×C₆₇	—	2	1	no	—
19	C₂×#117×C₆₇	—	2	1	no	—
20	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
21	#423×C₆₇	—	2	4	no	—
22	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
23	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
24	C₂×#103×C₆₇	C₂²	1	8	yes	—
25	#384×C₆₇	C₂²	1	2	yes	—
26	#382×C₆₇	—	2	1	no	—
27	#380×C₆₇	—	2	1	no	—
28	#380×C₆₇	C₂²	1	2	yes	—
29	C₄×C₄⋊C₄×C₆₇	—	2	1	no	—
30	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
31	#340×C₆₇	C₂²	1	8	yes	—
32	C₂²×Q₈×C₆₇	—	2	1	no	—
33	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
34	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
35	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
36	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
37	C₂×M₁₆×C₆₇	—	2	4	no	—
38	C₂×M₁₆×C₆₇	—	4	4	no	—
39	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
40	C₄⋊Q₈×C₆₇	—	2	2	no	—
41	C₄⋊Q₈×C₆₇	—	2	2	no	—
42	C₄⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
43	C₄⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
44	#125×C₆₇	—	2	8	no	—
45	C₂²⋊Q₈×C₆₇	—	2	8	no	—
46	C₂²⋊Q₈×C₆₇	—	2	4	no	—
47	C₂²⋊Q₈×C₆₇	C₂×C₄	1	8	yes	—
48	C₄×Q₈×C₆₇	—	2	8	no	—
49	C₄×Q₈×C₆₇	—	2	4	no	—
50	C₄×Q₈×C₆₇	C₂×C₄	1	8	yes	—
51	#117×C₆₇	—	2	8	no	—
52	#117×C₆₇	—	4	4	no	—
53	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
54	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
55	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
56	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
57	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
58	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
59	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
60	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
61	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
62	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
63	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
64	C₂×C₄²×C₆₇	—	2	2	no	—
65	C₂×C₄²×C₆₇	—	2	2	no	—
66	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
67	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
68	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
69	C₂×C₄²×C₆₇	D₄	1	4	yes	—
70	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
71	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
72	C₄⋊C₈×C₆₇	—	2	4	no	—
73	C₄⋊C₈×C₆₇	—	2	4	no	—
74	#103×C₆₇	—	4	8	no	—
75	#95×C₆₇	—	2	2	no	—
76	#95×C₆₇	—	2	2	no	—
77	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
78	C₂×Q₈×C₆₇	—	2	4	no	—
79	C₂×Q₈×C₆₇	—	2	2	no	—
80	C₂×Q₈×C₆₇	—	2	4	no	—
81	C₂×Q₈×C₆₇	—	2	4	no	—
82	C₂×Q₈×C₆₇	—	4	2	no	—
83	C₂×Q₈×C₆₇	—	4	2	no	—
84	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
85	C₂²×C₄×C₆₇	—	2	2	no	—
86	C₂²×C₄×C₆₇	—	2	2	no	—
87	C₂²×C₄×C₆₇	—	2	4	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	—	2	2	no	—
92	C₂²×C₄×C₆₇	—	2	4	no	—
93	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
94	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
95	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
96	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
99	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
100	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
103	M₁₆×C₆₇	—	4	8	no	—
104	C₂×C₈×C₆₇	—	2	4	no	—
105	C₂×C₈×C₆₇	—	2	8	no	—
106	C₄⋊C₄×C₆₇	—	2	4	no	—
107	C₄⋊C₄×C₆₇	—	2	4	no	—
108	C₄⋊C₄×C₆₇	—	2	4	no	—
109	C₄⋊C₄×C₆₇	—	2	8	no	—
110	C₄⋊C₄×C₆₇	—	2	8	no	—
111	C₄⋊C₄×C₆₇	—	2	2	no	—
112	C₄⋊C₄×C₆₇	—	2	4	no	—
113	C₄⋊C₄×C₆₇	—	2	4	no	—
114	C₄⋊C₄×C₆₇	—	4	4	no	—
115	C₄⋊C₄×C₆₇	—	4	4	no	—
116	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
117	C₂²⋊C₄×C₆₇	—	2	8	no	—
118	C₂²⋊C₄×C₆₇	—	4	4	no	—
119	C₄²×C₆₇	—	2	8	no	—
120	C₄²×C₆₇	—	4	4	no	—
121	C₄²×C₆₇	C₂²⋊C₄	1	4	yes	—
122	C₄²×C₆₇	C₂²⋊C₄	1	4	yes	—
123	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
124	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
125	C₄²×C₆₇	C₂×D₄	1	2	yes	—
126	C₄²×C₆₇	C₂×D₄	1	2	yes	—
127	C₂³×C₆₇	—	2	2	no	—
128	C₂³×C₆₇	—	2	1	no	—
129	C₂³×C₆₇	—	2	2	no	—
130	C₂³×C₆₇	—	2	1	no	—
131	C₂³×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
132	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
133	Q₈×C₆₇	—	2	8	no	—
134	Q₈×C₆₇	—	4	2	no	—
135	Q₈×C₆₇	—	4	2	no	—
136	Q₈×C₆₇	—	4	8	no	—
137	C₂×C₄×C₆₇	—	2	2	no	—
138	C₂×C₄×C₆₇	—	2	1	no	—
139	C₂×C₄×C₆₇	—	2	2	no	—
140	C₂×C₄×C₆₇	—	2	1	no	—
141	C₂×C₄×C₆₇	—	2	4	no	—
142	C₂×C₄×C₆₇	—	2	2	no	—
143	C₂×C₄×C₆₇	—	2	2	no	—
144	C₂×C₄×C₆₇	—	2	8	no	—
145	C₂×C₄×C₆₇	—	2	8	no	—
146	C₂×C₄×C₆₇	—	2	4	no	—
147	C₂×C₄×C₆₇	—	2	4	no	—
148	C₂×C₄×C₆₇	—	4	4	no	—
149	C₂×C₄×C₆₇	—	4	2	no	—
150	C₂×C₄×C₆₇	—	4	2	no	—
151	C₂×C₄×C₆₇	—	4	4	no	—
152	C₂×C₄×C₆₇	—	4	4	no	—
153	C₂×C₄×C₆₇	—	4	4	no	—
154	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
155	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
156	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
157	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
158	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
159	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
160	C₂×C₄×C₆₇	#137	1	2	yes	—
161	C₂×C₄×C₆₇	#137	1	4	yes	—
162	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
163	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
164	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
165	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
166	C₈×C₆₇	—	4	4	no	—
167	C₂²×C₆₇	—	2	2	no	—
168	C₂²×C₆₇	—	2	2	no	—
169	C₂²×C₆₇	—	2	2	no	—
170	C₂²×C₆₇	—	4	2	no	—
171	C₂²×C₆₇	—	4	2	no	—
172	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
173	C₂²×C₆₇	#537	1	2	yes	—
174	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
175	C₂²×C₆₇	C₂×#137	1	2	yes	—
176	C₂²×C₆₇	C₂×#137	1	1	yes	characteristic
177	C₄×C₆₇	—	2	2	no	—
178	C₄×C₆₇	—	2	2	no	—
179	C₄×C₆₇	—	2	2	no	—
180	C₄×C₆₇	—	4	4	no	—
181	C₄×C₆₇	—	4	4	no	—
182	C₄×C₆₇	—	8	4	no	—
183	C₄×C₆₇	#417	1	4	yes	—
184	C₄×C₆₇	#419	1	4	yes	—
185	C₄×C₆₇	#520	1	2	yes	—
186	C₄×C₆₇	#535	1	1	yes	characteristic
187	C₄×C₆₇	#536	1	1	yes	characteristic
188	#21894	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
189	C₂×C₆₇	—	4	2	no	—
190	C₂×C₆₇	#2311	1	1	yes	characteristic
191	C₂×C₆₇	C₂×#419	1	1	yes	characteristic
192	C₂×C₆₇	C₂×#417	1	1	yes	characteristic
193	C₂×C₆₇	#3254	1	2	yes	—
194	C₂×C₆₇	#3255	1	2	yes	—
195	C₂×#533	C₂×C₆₇	1	1	yes	characteristic
196	C₂×#423	C₂×C₆₇	1	1	yes	characteristic
197	#2285	C₂×C₆₇	1	1	yes	characteristic
198	#1988	C₂×C₆₇	1	2	yes	complemented
199	#1981	C₂×C₆₇	1	2	yes	complemented
200	#1894	C₂×C₆₇	1	4	yes	—
201	#1803	C₂×C₆₇	1	4	yes	complemented
202	C₆₇	#21894	1	1	yes	characteristic, direct product factor, Sylow
203	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
204	#533	—	2	4	no	—
205	#533	C₄×C₆₇	1	8	yes	complemented
206	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
207	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
208	C₂×#125	—	2	1	no	—
209	C₂×C₂²⋊Q₈	—	2	1	no	—
210	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
211	C₂×C₄×Q₈	—	2	1	no	—
212	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
213	C₂×#117	—	2	1	no	—
214	C₂×#117	—	2	1	no	—
215	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
216	#423	—	2	4	no	—
217	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
218	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
219	C₂×#103	C₂²×C₆₇	1	8	yes	—
220	#384	C₂²×C₆₇	1	2	yes	—
221	#382	—	2	1	no	—
222	#380	—	2	1	no	—
223	#380	C₂²×C₆₇	1	2	yes	—
224	C₄×C₄⋊C₄	—	2	1	no	—
225	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
226	#340	C₂²×C₆₇	1	8	yes	—
227	C₂²×Q₈	—	2	1	no	—
228	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
229	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
230	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
231	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
232	C₂×M₁₆	—	2	4	no	—
233	C₂×M₁₆	—	4	4	no	—
234	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
235	C₄⋊Q₈	—	2	2	no	—
236	C₄⋊Q₈	—	2	2	no	—
237	C₄⋊Q₈	C₂×C₄×C₆₇	1	4	yes	—
238	C₄⋊Q₈	C₂×C₄×C₆₇	1	4	yes	—
239	#125	—	2	8	no	—
240	C₂²⋊Q₈	—	2	8	no	—
241	C₂²⋊Q₈	—	2	4	no	—
242	C₂²⋊Q₈	C₂×C₄×C₆₇	1	8	yes	—
243	C₄×Q₈	—	2	8	no	—
244	C₄×Q₈	—	2	4	no	—
245	C₄×Q₈	C₂×C₄×C₆₇	1	8	yes	—
246	#117	—	2	8	no	—
247	#117	—	4	4	no	—
248	C₂×C₄⋊C₄	—	2	1	no	—
249	C₂×C₄⋊C₄	—	2	1	no	—
250	C₂×C₄⋊C₄	—	2	1	no	—
251	C₂×C₄⋊C₄	—	2	2	no	—
252	C₂×C₄⋊C₄	—	2	2	no	—
253	C₂×C₄⋊C₄	—	2	2	no	—
254	C₂×C₄⋊C₄	—	2	2	no	—
255	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
256	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
257	C₂×C₂²⋊C₄	—	2	2	no	—
258	C₂×C₂²⋊C₄	—	2	2	no	—
259	C₂×C₄²	—	2	2	no	—
260	C₂×C₄²	—	2	2	no	—
261	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
262	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
263	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
264	C₂×C₄²	D₄×C₆₇	1	4	yes	—
265	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
266	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
267	C₄⋊C₈	—	2	4	no	—
268	C₄⋊C₈	—	2	4	no	—
269	#103	—	4	8	no	—
270	#95	—	2	2	no	—
271	#95	—	2	2	no	—
272	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
273	C₂×Q₈	—	2	4	no	—
274	C₂×Q₈	—	2	2	no	—
275	C₂×Q₈	—	2	4	no	—
276	C₂×Q₈	—	2	4	no	—
277	C₂×Q₈	—	4	2	no	—
278	C₂×Q₈	—	4	2	no	—
279	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
280	C₂²×C₄	—	2	2	no	—
281	C₂²×C₄	—	2	2	no	—
282	C₂²×C₄	—	2	4	no	—
283	C₂²×C₄	—	2	2	no	—
284	C₂²×C₄	—	2	2	no	—
285	C₂²×C₄	—	2	2	no	—
286	C₂²×C₄	—	2	2	no	—
287	C₂²×C₄	—	2	4	no	—
288	C₂²×C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
289	C₂²×C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
290	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
291	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
292	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
293	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
294	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
295	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
296	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
297	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
298	M₁₆	—	4	8	no	—
299	C₂×C₈	—	2	4	no	—
300	C₂×C₈	—	2	8	no	—
301	C₄⋊C₄	—	2	4	no	—
302	C₄⋊C₄	—	2	4	no	—
303	C₄⋊C₄	—	2	4	no	—
304	C₄⋊C₄	—	2	8	no	—
305	C₄⋊C₄	—	2	8	no	—
306	C₄⋊C₄	—	2	2	no	—
307	C₄⋊C₄	—	2	4	no	—
308	C₄⋊C₄	—	2	4	no	—
309	C₄⋊C₄	—	4	4	no	—
310	C₄⋊C₄	—	4	4	no	—
311	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
312	C₂²⋊C₄	—	2	8	no	—
313	C₂²⋊C₄	—	4	4	no	—
314	C₄²	—	2	8	no	—
315	C₄²	—	4	4	no	—
316	C₄²	C₂²⋊C₄×C₆₇	1	4	yes	—
317	C₄²	C₂²⋊C₄×C₆₇	1	4	yes	—
318	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
319	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
320	C₄²	C₂×D₄×C₆₇	1	2	yes	—
321	C₄²	C₂×D₄×C₆₇	1	2	yes	—
322	C₂³	—	2	2	no	—
323	C₂³	—	2	1	no	—
324	C₂³	—	2	2	no	—
325	C₂³	—	2	1	no	—
326	C₂³	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
327	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
328	Q₈	—	2	8	no	—
329	Q₈	—	4	2	no	—
330	Q₈	—	4	2	no	—
331	Q₈	—	4	8	no	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	2	1	no	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	2	1	no	—
336	C₂×C₄	—	2	4	no	—
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	2	8	no	—
340	C₂×C₄	—	2	8	no	—
341	C₂×C₄	—	2	4	no	—
342	C₂×C₄	—	2	4	no	—
343	C₂×C₄	—	4	4	no	—
344	C₂×C₄	—	4	2	no	—
345	C₂×C₄	—	4	2	no	—
346	C₂×C₄	—	4	4	no	—
347	C₂×C₄	—	4	4	no	—
348	C₂×C₄	—	4	4	no	—
349	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
350	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
351	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
352	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
353	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
354	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
355	C₂×C₄	#137×C₆₇	1	2	yes	—
356	C₂×C₄	#137×C₆₇	1	4	yes	—
357	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
358	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
359	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
360	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
361	C₈	—	4	4	no	—
362	C₂²	—	2	2	no	—
363	C₂²	—	2	2	no	—
364	C₂²	—	2	2	no	—
365	C₂²	—	4	2	no	—
366	C₂²	—	4	2	no	—
367	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
368	C₂²	#537×C₆₇	1	2	yes	—
369	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
370	C₂²	C₂×#137×C₆₇	1	2	yes	—
371	C₂²	C₂×#137×C₆₇	1	1	yes	characteristic
372	C₄	—	2	2	no	—
373	C₄	—	2	2	no	—
374	C₄	—	2	2	no	—
375	C₄	—	4	4	no	—
376	C₄	—	4	4	no	—
377	C₄	—	8	4	no	—
378	C₄	#417×C₆₇	1	4	yes	—
379	C₄	#419×C₆₇	1	4	yes	—
380	C₄	#520×C₆₇	1	2	yes	—
381	C₄	#535×C₆₇	1	1	yes	characteristic
382	C₄	#536×C₆₇	1	1	yes	characteristic
383	C₂	—	4	2	no	—
384	C₂	C₆₇×#2311	1	1	yes	characteristic
385	C₂	C₂×#419×C₆₇	1	1	yes	characteristic
386	C₂	C₂×#417×C₆₇	1	1	yes	characteristic
387	C₂	C₆₇×#3254	1	2	yes	—
388	C₂	C₆₇×#3255	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	C₂⁴×C₆₇	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₆₇×#1988	C₂	semidirect
5. C₆₇×#1981	C₂	semidirect
7. C₆₇×#1803	C₂	semidirect
10. #533×C₆₇	C₄	semidirect
188. #21894	C₆₇	direct (N × H)
198. #1988	C₂×C₆₇	semidirect
199. #1981	C₂×C₆₇	semidirect
201. #1803	C₂×C₆₇	semidirect
202. C₆₇	#21894	direct (N × H)
205. #533	C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#533×C₆₇	2
C₂×#423×C₆₇	2
C₆₇×#2285	2
C₆₇×#1988	2
C₆₇×#1981	2
C₆₇×#1894	2
C₆₇×#1803	2
#21894	67

Subgroups — C67×#21894 #30701145

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21894 #30701145