Subgroups — C₆₇×#21393 #30700644

Order 17152.

1430 conjugacy classes of subgroups.

The subgroup lattice has 388 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×#432×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#117×C₆₇	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×#1875	C₂	1	4	yes	maximal, complemented
7	C₆₇×#1762	C₂	1	4	yes	maximal, complemented
8	C₂²×#41×C₆₇	—	2	1	no	—
9	#532×C₆₇	—	2	4	no	—
10	#532×C₆₇	C₄	1	8	yes	complemented
11	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×#124×C₆₇	—	2	1	no	—
14	C₂×#121×C₆₇	—	2	1	no	—
15	C₂×#121×C₆₇	C₂²	1	2	yes	—
16	C₂×C₄×D₄×C₆₇	—	2	1	no	—
17	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
18	C₂×#117×C₆₇	C₂²	1	2	yes	—
19	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
20	#432×C₆₇	—	2	4	no	—
21	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
22	C₂×#102×C₆₇	C₂²	1	8	yes	—
23	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
24	C₂×#97×C₆₇	C₂²	1	2	yes	—
25	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
26	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
27	C₄×C₂²⋊C₄×C₆₇	—	2	1	no	—
28	C₄×C₂²⋊C₄×C₆₇	—	2	1	no	—
29	#376×C₆₇	—	2	1	no	—
30	C₄³×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	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
44	C₄⋊Q₈×C₆₇	—	2	2	no	—
45	C₄⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
46	#127×C₆₇	—	2	2	no	—
47	#127×C₆₇	C₂×C₄	1	4	yes	—
48	#124×C₆₇	—	2	8	no	—
49	#121×C₆₇	—	2	8	no	—
50	#121×C₆₇	—	2	4	no	—
51	#121×C₆₇	C₂×C₄	1	8	yes	—
52	C₄×D₄×C₆₇	—	2	8	no	—
53	C₄×D₄×C₆₇	—	2	4	no	—
54	C₄×D₄×C₆₇	C₂×C₄	1	8	yes	—
55	#117×C₆₇	—	2	8	no	—
56	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
57	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
58	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
59	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
60	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
61	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
62	C₂×C₄²×C₆₇	—	2	2	no	—
63	C₂×C₄²×C₆₇	—	2	2	no	—
64	C₂×C₄²×C₆₇	—	2	2	no	—
65	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
66	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
67	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
68	C₂×C₄²×C₆₇	D₄	1	4	yes	—
69	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
70	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
71	C₄⋊C₈×C₆₇	—	2	4	no	—
72	#102×C₆₇	—	2	16	no	—
73	C₂²⋊C₈×C₆₇	—	2	4	no	—
74	#97×C₆₇	—	2	4	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	8	no	—
87	C₂×D₄×C₆₇	—	4	4	no	—
88	C₂×D₄×C₆₇	—	4	8	no	—
89	C₂×D₄×C₆₇	—	4	4	no	—
90	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
91	C₂²×C₄×C₆₇	—	2	4	no	—
92	C₂²×C₄×C₆₇	—	2	4	no	—
93	C₂²×C₄×C₆₇	—	2	4	no	—
94	C₂²×C₄×C₆₇	—	2	2	no	—
95	C₂²×C₄×C₆₇	—	2	8	no	—
96	C₂²×C₄×C₆₇	—	4	2	no	—
97	C₂²×C₄×C₆₇	—	4	8	no	—
98	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
99	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	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×D₄	1	1	yes	characteristic
105	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
106	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
108	C₂×C₈×C₆₇	—	2	4	no	—
109	C₂×C₈×C₆₇	—	2	8	no	—
110	C₄⋊C₄×C₆₇	—	2	4	no	—
111	C₄⋊C₄×C₆₇	—	2	2	no	—
112	C₄⋊C₄×C₆₇	—	2	4	no	—
113	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
114	C₄⋊C₄×C₆₇	#41	1	8	yes	complemented
115	C₂²⋊C₄×C₆₇	—	2	8	no	—
116	C₂²⋊C₄×C₆₇	—	2	8	no	—
117	C₂²⋊C₄×C₆₇	—	2	8	no	—
118	C₄²×C₆₇	—	2	8	no	—
119	C₄²×C₆₇	—	4	4	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	2	no	—
129	C₂³×C₆₇	—	2	2	no	—
130	C₂³×C₆₇	—	2	4	no	—
131	C₂³×C₆₇	—	4	4	no	—
132	C₂³×C₆₇	—	4	4	no	—
133	C₂³×C₆₇	—	4	4	no	—
134	C₂³×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
135	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
136	Q₈×C₆₇	—	4	8	no	—
137	D₄×C₆₇	—	2	8	no	—
138	D₄×C₆₇	—	4	8	no	—
139	D₄×C₆₇	—	4	16	no	—
140	D₄×C₆₇	—	4	4	no	—
141	D₄×C₆₇	—	4	8	no	—
142	C₂×C₄×C₆₇	—	2	2	no	—
143	C₂×C₄×C₆₇	—	2	2	no	—
144	C₂×C₄×C₆₇	—	2	2	no	—
145	C₂×C₄×C₆₇	—	2	2	no	—
146	C₂×C₄×C₆₇	—	2	4	no	—
147	C₂×C₄×C₆₇	—	2	4	no	—
148	C₂×C₄×C₆₇	—	2	8	no	—
149	C₂×C₄×C₆₇	—	2	4	no	—
150	C₂×C₄×C₆₇	—	2	8	no	—
151	C₂×C₄×C₆₇	—	4	4	no	—
152	C₂×C₄×C₆₇	—	4	16	no	—
153	C₂×C₄×C₆₇	—	4	4	no	—
154	C₂×C₄×C₆₇	—	4	4	no	—
155	C₂×C₄×C₆₇	—	4	4	no	—
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	C₂×C₂²⋊C₄	1	2	yes	—
160	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
161	C₂×C₄×C₆₇	AGL(1,ℤ/8)	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	C₂×#41	1	4	yes	—
165	C₈×C₆₇	—	4	4	no	—
166	C₂²×C₆₇	—	2	2	no	—
167	C₂²×C₆₇	—	2	2	no	—
168	C₂²×C₆₇	—	2	2	no	—
169	C₂²×C₆₇	—	4	4	no	—
170	C₂²×C₆₇	—	4	4	no	—
171	C₂²×C₆₇	—	4	4	no	—
172	C₂²×C₆₇	—	4	4	no	—
173	C₂²×C₆₇	—	8	8	no	—
174	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
175	C₂²×C₆₇	#525	1	2	yes	—
176	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
177	C₂²×C₆₇	#575	1	2	yes	—
178	C₄×C₆₇	—	2	2	no	—
179	C₄×C₆₇	—	2	2	no	—
180	C₄×C₆₇	—	2	2	no	—
181	C₄×C₆₇	—	2	2	no	—
182	C₄×C₆₇	—	4	8	no	—
183	C₄×C₆₇	—	4	4	no	—
184	C₄×C₆₇	#418	1	4	yes	—
185	C₄×C₆₇	C₄×#41	1	2	yes	—
186	C₄×C₆₇	#532	1	2	yes	—
187	#21393	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
188	C₂×C₆₇	—	4	2	no	—
189	C₂×C₆₇	—	8	4	no	—
190	C₂×C₆₇	#2297	1	1	yes	characteristic
191	C₂×C₆₇	C₂×#418	1	1	yes	characteristic
192	C₂×C₆₇	#2896	1	1	yes	characteristic
193	C₂×C₆₇	#3115	1	4	yes	—
194	C₂×#532	C₂×C₆₇	1	1	yes	characteristic
195	C₂×#432	C₂×C₆₇	1	1	yes	characteristic, complemented
196	C₄×#117	C₂×C₆₇	1	1	yes	characteristic, complemented
197	#1988	C₂×C₆₇	1	2	yes	complemented
198	#1982	C₂×C₆₇	1	2	yes	complemented
199	#1875	C₂×C₆₇	1	4	yes	complemented
200	#1762	C₂×C₆₇	1	4	yes	complemented
201	C₆₇	#21393	1	1	yes	characteristic, direct product factor, Sylow
202	C₂²×#41	—	2	1	no	—
203	#532	—	2	4	no	—
204	#532	C₄×C₆₇	1	8	yes	complemented
205	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
206	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
207	C₂×#124	—	2	1	no	—
208	C₂×#121	—	2	1	no	—
209	C₂×#121	C₂²×C₆₇	1	2	yes	—
210	C₂×C₄×D₄	—	2	1	no	—
211	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
212	C₂×#117	C₂²×C₆₇	1	2	yes	—
213	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
214	#432	—	2	4	no	—
215	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
216	C₂×#102	C₂²×C₆₇	1	8	yes	—
217	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
218	C₂×#97	C₂²×C₆₇	1	2	yes	—
219	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
220	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
221	C₄×C₂²⋊C₄	—	2	1	no	—
222	C₄×C₂²⋊C₄	—	2	1	no	—
223	#376	—	2	1	no	—
224	C₄³	—	2	1	no	—
225	#340	C₂²×C₆₇	1	8	yes	—
226	C₂×#41	—	2	8	no	—
227	C₂×#41	—	4	8	no	—
228	C₂²×Q₈	—	2	1	no	—
229	C₂²×D₄	—	2	2	no	—
230	C₂²×D₄	—	2	1	no	—
231	C₂²×D₄	—	2	1	no	—
232	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
233	C₂³×C₄	—	2	2	no	—
234	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
235	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
236	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
237	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
238	C₄⋊Q₈	—	2	2	no	—
239	C₄⋊Q₈	C₂×C₄×C₆₇	1	4	yes	—
240	#127	—	2	2	no	—
241	#127	C₂×C₄×C₆₇	1	4	yes	—
242	#124	—	2	8	no	—
243	#121	—	2	8	no	—
244	#121	—	2	4	no	—
245	#121	C₂×C₄×C₆₇	1	8	yes	—
246	C₄×D₄	—	2	8	no	—
247	C₄×D₄	—	2	4	no	—
248	C₄×D₄	C₂×C₄×C₆₇	1	8	yes	—
249	#117	—	2	8	no	—
250	C₂×C₄⋊C₄	—	2	1	no	—
251	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
252	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
253	C₂×C₂²⋊C₄	—	2	2	no	—
254	C₂×C₂²⋊C₄	—	2	2	no	—
255	C₂×C₂²⋊C₄	—	2	2	no	—
256	C₂×C₄²	—	2	2	no	—
257	C₂×C₄²	—	2	2	no	—
258	C₂×C₄²	—	2	2	no	—
259	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
260	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
261	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
262	C₂×C₄²	D₄×C₆₇	1	4	yes	—
263	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
264	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
265	C₄⋊C₈	—	2	4	no	—
266	#102	—	2	16	no	—
267	C₂²⋊C₈	—	2	4	no	—
268	#97	—	2	4	no	—
269	#95	—	2	2	no	—
270	#95	—	2	2	no	—
271	C₂⁴	—	2	2	no	—
272	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
273	#41	—	4	32	no	—
274	C₂×Q₈	—	2	4	no	—
275	C₂×Q₈	—	4	4	no	—
276	C₂×D₄	—	2	8	no	—
277	C₂×D₄	—	2	4	no	—
278	C₂×D₄	—	2	4	no	—
279	C₂×D₄	—	2	2	no	—
280	C₂×D₄	—	2	8	no	—
281	C₂×D₄	—	4	4	no	—
282	C₂×D₄	—	4	8	no	—
283	C₂×D₄	—	4	4	no	—
284	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
285	C₂²×C₄	—	2	4	no	—
286	C₂²×C₄	—	2	4	no	—
287	C₂²×C₄	—	2	4	no	—
288	C₂²×C₄	—	2	2	no	—
289	C₂²×C₄	—	2	8	no	—
290	C₂²×C₄	—	4	2	no	—
291	C₂²×C₄	—	4	8	no	—
292	C₂²×C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
293	C₂²×C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
294	C₂²×C₄	C₂²×C₄×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	C₂×D₄×C₆₇	1	1	yes	characteristic
298	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
299	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
300	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
301	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
302	C₂×C₈	—	2	4	no	—
303	C₂×C₈	—	2	8	no	—
304	C₄⋊C₄	—	2	4	no	—
305	C₄⋊C₄	—	2	2	no	—
306	C₄⋊C₄	—	2	4	no	—
307	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
308	C₄⋊C₄	#41×C₆₇	1	8	yes	complemented
309	C₂²⋊C₄	—	2	8	no	—
310	C₂²⋊C₄	—	2	8	no	—
311	C₂²⋊C₄	—	2	8	no	—
312	C₄²	—	2	8	no	—
313	C₄²	—	4	4	no	—
314	C₄²	—	4	4	no	—
315	C₄²	C₂²⋊C₄×C₆₇	1	4	yes	—
316	C₄²	C₂²⋊C₄×C₆₇	1	4	yes	—
317	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
318	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
319	C₄²	C₂×D₄×C₆₇	1	2	yes	—
320	C₄²	C₂×D₄×C₆₇	1	2	yes	—
321	C₂³	—	2	2	no	—
322	C₂³	—	2	2	no	—
323	C₂³	—	2	2	no	—
324	C₂³	—	2	4	no	—
325	C₂³	—	4	4	no	—
326	C₂³	—	4	4	no	—
327	C₂³	—	4	4	no	—
328	C₂³	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
329	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
330	Q₈	—	4	8	no	—
331	D₄	—	2	8	no	—
332	D₄	—	4	8	no	—
333	D₄	—	4	16	no	—
334	D₄	—	4	4	no	—
335	D₄	—	4	8	no	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	2	2	no	—
340	C₂×C₄	—	2	4	no	—
341	C₂×C₄	—	2	4	no	—
342	C₂×C₄	—	2	8	no	—
343	C₂×C₄	—	2	4	no	—
344	C₂×C₄	—	2	8	no	—
345	C₂×C₄	—	4	4	no	—
346	C₂×C₄	—	4	16	no	—
347	C₂×C₄	—	4	4	no	—
348	C₂×C₄	—	4	4	no	—
349	C₂×C₄	—	4	4	no	—
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	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
355	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	4	yes	—
356	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
357	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
358	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
359	C₈	—	4	4	no	—
360	C₂²	—	2	2	no	—
361	C₂²	—	2	2	no	—
362	C₂²	—	2	2	no	—
363	C₂²	—	4	4	no	—
364	C₂²	—	4	4	no	—
365	C₂²	—	4	4	no	—
366	C₂²	—	4	4	no	—
367	C₂²	—	8	8	no	—
368	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
369	C₂²	#525×C₆₇	1	2	yes	—
370	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	2	yes	—
371	C₂²	#575×C₆₇	1	2	yes	—
372	C₄	—	2	2	no	—
373	C₄	—	2	2	no	—
374	C₄	—	2	2	no	—
375	C₄	—	2	2	no	—
376	C₄	—	4	8	no	—
377	C₄	—	4	4	no	—
378	C₄	#418×C₆₇	1	4	yes	—
379	C₄	C₄×#41×C₆₇	1	2	yes	—
380	C₄	#532×C₆₇	1	2	yes	—
381	C₂	—	4	2	no	—
382	C₂	—	8	4	no	—
383	C₂	C₆₇×#2297	1	1	yes	characteristic
384	C₂	C₂×#418×C₆₇	1	1	yes	characteristic
385	C₂	C₆₇×#2896	1	1	yes	characteristic
386	C₂	C₆₇×#3115	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×#432×C₆₇	C₂	semidirect
3. C₄×#117×C₆₇	C₂	semidirect
4. C₆₇×#1988	C₂	semidirect
5. C₆₇×#1982	C₂	semidirect
6. C₆₇×#1875	C₂	semidirect
7. C₆₇×#1762	C₂	semidirect
10. #532×C₆₇	C₄	semidirect
114. C₄⋊C₄×C₆₇	#41	semidirect
187. #21393	C₆₇	direct (N × H)
195. C₂×#432	C₂×C₆₇	semidirect
196. C₄×#117	C₂×C₆₇	semidirect
197. #1988	C₂×C₆₇	semidirect
198. #1982	C₂×C₆₇	semidirect
199. #1875	C₂×C₆₇	semidirect
200. #1762	C₂×C₆₇	semidirect
201. C₆₇	#21393	direct (N × H)
204. #532	C₄×C₆₇	semidirect
308. C₄⋊C₄	#41×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×#432×C₆₇	2
C₄×#117×C₆₇	2
C₆₇×#1988	2
C₆₇×#1982	2
C₆₇×#1875	2
C₆₇×#1762	2
#21393	67

Subgroups — C67×#21393 #30700644

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21393 #30700644