Subgroups — C₆₇×#21891 #30701142

Order 17152.

1366 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#532×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#423×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2283	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1988	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1982	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1895	C₂	1	4	yes	maximal, complemented
7	C₆₇×#1804	C₂	1	4	yes	maximal, complemented
8	C₂²×#41×C₆₇	—	2	1	no	—
9	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
10	#532×C₆₇	—	2	4	no	—
11	#532×C₆₇	C₄	1	8	yes	complemented
12	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×#124×C₆₇	—	2	1	no	—
15	C₂×#121×C₆₇	—	2	1	no	—
16	C₂×#121×C₆₇	C₂²	1	2	yes	—
17	C₂×C₄×D₄×C₆₇	—	2	1	no	—
18	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
19	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
20	#423×C₆₇	—	2	4	no	—
21	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
22	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
23	C₂×#102×C₆₇	C₂²	1	8	yes	—
24	#383×C₆₇	—	2	1	no	—
25	#382×C₆₇	C₂²	1	2	yes	—
26	#381×C₆₇	—	2	1	no	—
27	#380×C₆₇	C₂²	1	2	yes	—
28	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
29	C₄×C₂²⋊C₄×C₆₇	—	2	1	no	—
30	#376×C₆₇	—	2	1	no	—
31	#340×C₆₇	C₂²	1	8	yes	—
32	C₂×#41×C₆₇	—	2	8	no	—
33	C₂×#41×C₆₇	—	4	8	no	—
34	C₂²×Q₈×C₆₇	—	2	1	no	—
35	C₂²×D₄×C₆₇	—	2	2	no	—
36	C₂²×D₄×C₆₇	—	2	1	no	—
37	C₂²×D₄×C₆₇	—	2	1	no	—
38	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
39	C₂³×C₄×C₆₇	—	2	2	no	—
40	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
41	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
42	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
43	C₂×M₁₆×C₆₇	—	2	4	no	—
44	C₂×M₁₆×C₆₇	—	4	4	no	—
45	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
46	C₄⋊Q₈×C₆₇	—	2	2	no	—
47	C₄⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
48	#127×C₆₇	—	2	2	no	—
49	#127×C₆₇	C₂×C₄	1	4	yes	—
50	#124×C₆₇	—	2	8	no	—
51	#121×C₆₇	—	2	8	no	—
52	#121×C₆₇	—	2	4	no	—
53	#121×C₆₇	C₂×C₄	1	8	yes	—
54	C₄×D₄×C₆₇	—	2	8	no	—
55	C₄×D₄×C₆₇	—	2	4	no	—
56	C₄×D₄×C₆₇	C₂×C₄	1	8	yes	—
57	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
58	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
59	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
60	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
61	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
62	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
63	C₂×C₄²×C₆₇	—	2	2	no	—
64	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
65	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
66	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
67	C₂×C₄²×C₆₇	D₄	1	4	yes	—
68	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
69	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
70	C₄⋊C₈×C₆₇	—	2	4	no	—
71	C₄⋊C₈×C₆₇	—	2	4	no	—
72	#102×C₆₇	—	4	8	no	—
73	#95×C₆₇	—	2	2	no	—
74	#95×C₆₇	—	2	2	no	—
75	#95×C₆₇	—	2	2	no	—
76	#95×C₆₇	—	2	2	no	—
77	C₂⁴×C₆₇	—	2	2	no	—
78	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
79	#41×C₆₇	—	4	32	no	—
80	C₂×Q₈×C₆₇	—	2	4	no	—
81	C₂×Q₈×C₆₇	—	4	4	no	—
82	C₂×D₄×C₆₇	—	2	8	no	—
83	C₂×D₄×C₆₇	—	2	4	no	—
84	C₂×D₄×C₆₇	—	2	4	no	—
85	C₂×D₄×C₆₇	—	2	2	no	—
86	C₂×D₄×C₆₇	—	2	4	no	—
87	C₂×D₄×C₆₇	—	2	4	no	—
88	C₂×D₄×C₆₇	—	4	4	no	—
89	C₂×D₄×C₆₇	—	4	8	no	—
90	C₂×D₄×C₆₇	—	4	4	no	—
91	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
92	C₂²×C₄×C₆₇	—	2	4	no	—
93	C₂²×C₄×C₆₇	—	2	2	no	—
94	C₂²×C₄×C₆₇	—	2	2	no	—
95	C₂²×C₄×C₆₇	—	2	2	no	—
96	C₂²×C₄×C₆₇	—	2	8	no	—
97	C₂²×C₄×C₆₇	—	2	4	no	—
98	C₂²×C₄×C₆₇	—	4	2	no	—
99	C₂²×C₄×C₆₇	—	4	8	no	—
100	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
101	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
102	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
103	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
108	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
110	M₁₆×C₆₇	—	4	8	no	—
111	C₂×C₈×C₆₇	—	2	4	no	—
112	C₂×C₈×C₆₇	—	2	8	no	—
113	C₄⋊C₄×C₆₇	—	2	4	no	—
114	C₄⋊C₄×C₆₇	—	2	2	no	—
115	C₄⋊C₄×C₆₇	—	2	4	no	—
116	C₄⋊C₄×C₆₇	—	2	4	no	—
117	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
118	C₂²⋊C₄×C₆₇	—	2	8	no	—
119	C₂²⋊C₄×C₆₇	—	2	8	no	—
120	C₂²⋊C₄×C₆₇	—	4	4	no	—
121	C₄²×C₆₇	—	4	4	no	—
122	C₄²×C₆₇	C₂²⋊C₄	1	4	yes	—
123	C₄²×C₆₇	C₂²⋊C₄	1	4	yes	—
124	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
125	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
126	C₄²×C₆₇	C₂×D₄	1	2	yes	—
127	C₄²×C₆₇	C₂×D₄	1	2	yes	—
128	C₂³×C₆₇	—	2	2	no	—
129	C₂³×C₆₇	—	2	2	no	—
130	C₂³×C₆₇	—	2	2	no	—
131	C₂³×C₆₇	—	2	4	no	—
132	C₂³×C₆₇	—	4	4	no	—
133	C₂³×C₆₇	—	4	4	no	—
134	C₂³×C₆₇	—	4	4	no	—
135	C₂³×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
136	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
137	Q₈×C₆₇	—	4	8	no	—
138	D₄×C₆₇	—	2	8	no	—
139	D₄×C₆₇	—	4	8	no	—
140	D₄×C₆₇	—	4	16	no	—
141	D₄×C₆₇	—	4	2	no	—
142	D₄×C₆₇	—	4	2	no	—
143	D₄×C₆₇	—	4	8	no	—
144	C₂×C₄×C₆₇	—	2	2	no	—
145	C₂×C₄×C₆₇	—	2	2	no	—
146	C₂×C₄×C₆₇	—	2	2	no	—
147	C₂×C₄×C₆₇	—	2	2	no	—
148	C₂×C₄×C₆₇	—	2	4	no	—
149	C₂×C₄×C₆₇	—	2	2	no	—
150	C₂×C₄×C₆₇	—	2	2	no	—
151	C₂×C₄×C₆₇	—	2	8	no	—
152	C₂×C₄×C₆₇	—	2	4	no	—
153	C₂×C₄×C₆₇	—	4	4	no	—
154	C₂×C₄×C₆₇	—	4	16	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	4	no	—
159	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
160	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
161	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
162	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
163	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
164	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
165	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
166	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
167	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
168	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
169	C₈×C₆₇	—	4	4	no	—
170	C₂²×C₆₇	—	2	2	no	—
171	C₂²×C₆₇	—	2	2	no	—
172	C₂²×C₆₇	—	2	2	no	—
173	C₂²×C₆₇	—	4	4	no	—
174	C₂²×C₆₇	—	4	4	no	—
175	C₂²×C₆₇	—	4	4	no	—
176	C₂²×C₆₇	—	4	4	no	—
177	C₂²×C₆₇	—	8	8	no	—
178	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
179	C₂²×C₆₇	#537	1	2	yes	—
180	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
181	C₂²×C₆₇	#575	1	2	yes	—
182	C₄×C₆₇	—	2	2	no	—
183	C₄×C₆₇	—	2	2	no	—
184	C₄×C₆₇	—	2	2	no	—
185	C₄×C₆₇	—	2	2	no	—
186	C₄×C₆₇	—	4	4	no	—
187	C₄×C₆₇	—	8	4	no	—
188	C₄×C₆₇	#418	1	4	yes	—
189	C₄×C₆₇	#520	1	2	yes	—
190	C₄×C₆₇	#535	1	1	yes	characteristic
191	C₄×C₆₇	#536	1	1	yes	characteristic
192	#21891	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
193	C₂×C₆₇	—	4	2	no	—
194	C₂×C₆₇	—	8	4	no	—
195	C₂×C₆₇	#2311	1	1	yes	characteristic
196	C₂×C₆₇	C₂×#418	1	1	yes	characteristic
197	C₂×C₆₇	#2896	1	1	yes	characteristic
198	C₂×C₆₇	#3251	1	4	yes	—
199	C₂×#532	C₂×C₆₇	1	1	yes	characteristic
200	C₂×#423	C₂×C₆₇	1	1	yes	characteristic, complemented
201	#2283	C₂×C₆₇	1	1	yes	characteristic, complemented
202	#1988	C₂×C₆₇	1	2	yes	complemented
203	#1982	C₂×C₆₇	1	2	yes	complemented
204	#1895	C₂×C₆₇	1	4	yes	complemented
205	#1804	C₂×C₆₇	1	4	yes	complemented
206	C₆₇	#21891	1	1	yes	characteristic, direct product factor, Sylow
207	C₂²×#41	—	2	1	no	—
208	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
209	#532	—	2	4	no	—
210	#532	C₄×C₆₇	1	8	yes	complemented
211	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
212	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
213	C₂×#124	—	2	1	no	—
214	C₂×#121	—	2	1	no	—
215	C₂×#121	C₂²×C₆₇	1	2	yes	—
216	C₂×C₄×D₄	—	2	1	no	—
217	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
218	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
219	#423	—	2	4	no	—
220	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
221	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
222	C₂×#102	C₂²×C₆₇	1	8	yes	—
223	#383	—	2	1	no	—
224	#382	C₂²×C₆₇	1	2	yes	—
225	#381	—	2	1	no	—
226	#380	C₂²×C₆₇	1	2	yes	—
227	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
228	C₄×C₂²⋊C₄	—	2	1	no	—
229	#376	—	2	1	no	—
230	#340	C₂²×C₆₇	1	8	yes	—
231	C₂×#41	—	2	8	no	—
232	C₂×#41	—	4	8	no	—
233	C₂²×Q₈	—	2	1	no	—
234	C₂²×D₄	—	2	2	no	—
235	C₂²×D₄	—	2	1	no	—
236	C₂²×D₄	—	2	1	no	—
237	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
238	C₂³×C₄	—	2	2	no	—
239	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
240	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
241	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
242	C₂×M₁₆	—	2	4	no	—
243	C₂×M₁₆	—	4	4	no	—
244	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
245	C₄⋊Q₈	—	2	2	no	—
246	C₄⋊Q₈	C₂×C₄×C₆₇	1	4	yes	—
247	#127	—	2	2	no	—
248	#127	C₂×C₄×C₆₇	1	4	yes	—
249	#124	—	2	8	no	—
250	#121	—	2	8	no	—
251	#121	—	2	4	no	—
252	#121	C₂×C₄×C₆₇	1	8	yes	—
253	C₄×D₄	—	2	8	no	—
254	C₄×D₄	—	2	4	no	—
255	C₄×D₄	C₂×C₄×C₆₇	1	8	yes	—
256	C₂×C₄⋊C₄	—	2	1	no	—
257	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
258	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
259	C₂×C₂²⋊C₄	—	2	2	no	—
260	C₂×C₂²⋊C₄	—	2	2	no	—
261	C₂×C₂²⋊C₄	—	2	2	no	—
262	C₂×C₄²	—	2	2	no	—
263	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
264	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
265	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
266	C₂×C₄²	D₄×C₆₇	1	4	yes	—
267	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
268	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
269	C₄⋊C₈	—	2	4	no	—
270	C₄⋊C₈	—	2	4	no	—
271	#102	—	4	8	no	—
272	#95	—	2	2	no	—
273	#95	—	2	2	no	—
274	#95	—	2	2	no	—
275	#95	—	2	2	no	—
276	C₂⁴	—	2	2	no	—
277	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
278	#41	—	4	32	no	—
279	C₂×Q₈	—	2	4	no	—
280	C₂×Q₈	—	4	4	no	—
281	C₂×D₄	—	2	8	no	—
282	C₂×D₄	—	2	4	no	—
283	C₂×D₄	—	2	4	no	—
284	C₂×D₄	—	2	2	no	—
285	C₂×D₄	—	2	4	no	—
286	C₂×D₄	—	2	4	no	—
287	C₂×D₄	—	4	4	no	—
288	C₂×D₄	—	4	8	no	—
289	C₂×D₄	—	4	4	no	—
290	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
291	C₂²×C₄	—	2	4	no	—
292	C₂²×C₄	—	2	2	no	—
293	C₂²×C₄	—	2	2	no	—
294	C₂²×C₄	—	2	2	no	—
295	C₂²×C₄	—	2	8	no	—
296	C₂²×C₄	—	2	4	no	—
297	C₂²×C₄	—	4	2	no	—
298	C₂²×C₄	—	4	8	no	—
299	C₂²×C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
300	C₂²×C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
301	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
302	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
303	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
304	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
305	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
306	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
307	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
308	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
309	M₁₆	—	4	8	no	—
310	C₂×C₈	—	2	4	no	—
311	C₂×C₈	—	2	8	no	—
312	C₄⋊C₄	—	2	4	no	—
313	C₄⋊C₄	—	2	2	no	—
314	C₄⋊C₄	—	2	4	no	—
315	C₄⋊C₄	—	2	4	no	—
316	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
317	C₂²⋊C₄	—	2	8	no	—
318	C₂²⋊C₄	—	2	8	no	—
319	C₂²⋊C₄	—	4	4	no	—
320	C₄²	—	4	4	no	—
321	C₄²	C₂²⋊C₄×C₆₇	1	4	yes	—
322	C₄²	C₂²⋊C₄×C₆₇	1	4	yes	—
323	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
324	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
325	C₄²	C₂×D₄×C₆₇	1	2	yes	—
326	C₄²	C₂×D₄×C₆₇	1	2	yes	—
327	C₂³	—	2	2	no	—
328	C₂³	—	2	2	no	—
329	C₂³	—	2	2	no	—
330	C₂³	—	2	4	no	—
331	C₂³	—	4	4	no	—
332	C₂³	—	4	4	no	—
333	C₂³	—	4	4	no	—
334	C₂³	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
335	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
336	Q₈	—	4	8	no	—
337	D₄	—	2	8	no	—
338	D₄	—	4	8	no	—
339	D₄	—	4	16	no	—
340	D₄	—	4	2	no	—
341	D₄	—	4	2	no	—
342	D₄	—	4	8	no	—
343	C₂×C₄	—	2	2	no	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	2	2	no	—
346	C₂×C₄	—	2	2	no	—
347	C₂×C₄	—	2	4	no	—
348	C₂×C₄	—	2	2	no	—
349	C₂×C₄	—	2	2	no	—
350	C₂×C₄	—	2	8	no	—
351	C₂×C₄	—	2	4	no	—
352	C₂×C₄	—	4	4	no	—
353	C₂×C₄	—	4	16	no	—
354	C₂×C₄	—	4	4	no	—
355	C₂×C₄	—	4	4	no	—
356	C₂×C₄	—	4	4	no	—
357	C₂×C₄	—	4	4	no	—
358	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
359	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
360	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
361	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
362	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
363	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	4	yes	—
364	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
365	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
366	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
367	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
368	C₈	—	4	4	no	—
369	C₂²	—	2	2	no	—
370	C₂²	—	2	2	no	—
371	C₂²	—	2	2	no	—
372	C₂²	—	4	4	no	—
373	C₂²	—	4	4	no	—
374	C₂²	—	4	4	no	—
375	C₂²	—	4	4	no	—
376	C₂²	—	8	8	no	—
377	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
378	C₂²	#537×C₆₇	1	2	yes	—
379	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	2	yes	—
380	C₂²	#575×C₆₇	1	2	yes	—
381	C₄	—	2	2	no	—
382	C₄	—	2	2	no	—
383	C₄	—	2	2	no	—
384	C₄	—	2	2	no	—
385	C₄	—	4	4	no	—
386	C₄	—	8	4	no	—
387	C₄	#418×C₆₇	1	4	yes	—
388	C₄	#520×C₆₇	1	2	yes	—
389	C₄	#535×C₆₇	1	1	yes	characteristic
390	C₄	#536×C₆₇	1	1	yes	characteristic
391	C₂	—	4	2	no	—
392	C₂	—	8	4	no	—
393	C₂	C₆₇×#2311	1	1	yes	characteristic
394	C₂	C₂×#418×C₆₇	1	1	yes	characteristic
395	C₂	C₆₇×#2896	1	1	yes	characteristic
396	C₂	C₆₇×#3251	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂×#532×C₆₇	C₂
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
2. C₂×#423×C₆₇	C₂	semidirect
3. C₆₇×#2283	C₂	semidirect
4. C₆₇×#1988	C₂	semidirect
5. C₆₇×#1982	C₂	semidirect
6. C₆₇×#1895	C₂	semidirect
7. C₆₇×#1804	C₂	semidirect
11. #532×C₆₇	C₄	semidirect
192. #21891	C₆₇	direct (N × H)
200. C₂×#423	C₂×C₆₇	semidirect
201. #2283	C₂×C₆₇	semidirect
202. #1988	C₂×C₆₇	semidirect
203. #1982	C₂×C₆₇	semidirect
204. #1895	C₂×C₆₇	semidirect
205. #1804	C₂×C₆₇	semidirect
206. C₆₇	#21891	direct (N × H)
210. #532	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×#532×C₆₇	2
C₂×#423×C₆₇	2
C₆₇×#2283	2
C₆₇×#1988	2
C₆₇×#1982	2
C₆₇×#1895	2
C₆₇×#1804	2
#21891	67

Subgroups — C67×#21891 #30701142

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21891 #30701142