Subgroups — C₆₇×#21838 #30701089

Order 17152.

1066 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#125×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#432×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2285	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1992	C₂	1	4	yes	maximal, complemented
5	C₆₇×#1989	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1824	C₂	1	2	yes	maximal
7	C₆₇×#1823	C₂	1	2	yes	maximal
8	C₂×#125×C₆₇	—	2	1	no	—
9	C₂×#125×C₆₇	—	2	1	no	—
10	C₂×#125×C₆₇	—	2	4	no	—
11	C₂×#125×C₆₇	C₄	1	8	yes	complemented
12	C₂×#125×C₆₇	C₂²	1	2	yes	—
13	C₂×#125×C₆₇	C₂²	1	2	yes	—
14	C₂×#117×C₆₇	—	2	1	no	—
15	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
16	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
17	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
18	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
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×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
23	C₂×#97×C₆₇	C₂²	1	2	yes	—
24	#384×C₆₇	C₂²	1	2	yes	—
25	#382×C₆₇	—	2	1	no	—
26	#380×C₆₇	—	2	1	no	—
27	#380×C₆₇	C₂²	1	2	yes	—
28	C₄×C₄⋊C₄×C₆₇	—	2	1	no	—
29	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
30	#340×C₆₇	C₂²	1	8	yes	—
31	#340×C₆₇	C₂²	1	8	yes	—
32	C₂³×C₄×C₆₇	—	2	1	no	—
33	C₂³×C₄×C₆₇	—	2	1	no	—
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	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
38	#125×C₆₇	—	2	8	no	—
39	#125×C₆₇	—	2	8	no	—
40	#125×C₆₇	—	2	4	no	—
41	#125×C₆₇	—	2	4	no	—
42	#125×C₆₇	C₂×C₄	1	8	yes	—
43	#125×C₆₇	C₂×C₄	1	8	yes	—
44	#117×C₆₇	—	4	4	no	—
45	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
46	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
47	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
48	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
49	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
50	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
51	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
52	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
53	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
54	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
55	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
56	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
57	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
58	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
59	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
60	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
61	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
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	C₂²⋊C₈×C₆₇	—	2	4	no	—
72	#97×C₆₇	—	2	4	no	—
73	#95×C₆₇	—	2	2	no	—
74	#95×C₆₇	—	2	2	no	—
75	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₆₇	—	2	2	no	—
77	C₂²×C₄×C₆₇	—	2	2	no	—
78	C₂²×C₄×C₆₇	—	2	4	no	—
79	C₂²×C₄×C₆₇	—	2	4	no	—
80	C₂²×C₄×C₆₇	—	2	4	no	—
81	C₂²×C₄×C₆₇	—	2	2	no	—
82	C₂²×C₄×C₆₇	—	2	2	no	—
83	C₂²×C₄×C₆₇	—	2	4	no	—
84	C₂²×C₄×C₆₇	—	4	2	no	—
85	C₂²×C₄×C₆₇	—	4	4	no	—
86	C₂²×C₄×C₆₇	—	4	2	no	—
87	C₂²×C₄×C₆₇	—	4	2	no	—
88	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
89	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
90	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
91	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
97	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
98	C₂×C₈×C₆₇	—	2	4	no	—
99	C₂×C₈×C₆₇	—	2	8	no	—
100	C₄⋊C₄×C₆₇	—	2	8	no	—
101	C₄⋊C₄×C₆₇	—	2	8	no	—
102	C₄⋊C₄×C₆₇	—	2	8	no	—
103	C₄⋊C₄×C₆₇	—	2	8	no	—
104	C₄⋊C₄×C₆₇	—	2	4	no	—
105	C₄⋊C₄×C₆₇	—	2	4	no	—
106	C₄⋊C₄×C₆₇	—	2	2	no	—
107	C₄⋊C₄×C₆₇	—	2	2	no	—
108	C₄⋊C₄×C₆₇	—	2	8	no	—
109	C₄⋊C₄×C₆₇	—	4	4	no	—
110	C₄⋊C₄×C₆₇	—	4	4	no	—
111	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
112	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
113	C₂²⋊C₄×C₆₇	—	4	4	no	—
114	C₄²×C₆₇	—	4	4	no	—
115	C₄²×C₆₇	C₂²⋊C₄	1	4	yes	—
116	C₄²×C₆₇	C₂²⋊C₄	1	4	yes	—
117	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
118	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
119	C₄²×C₆₇	C₂×D₄	1	2	yes	—
120	C₄²×C₆₇	C₂×D₄	1	2	yes	—
121	C₂³×C₆₇	—	2	2	no	—
122	C₂³×C₆₇	—	2	2	no	—
123	C₂³×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
124	C₂³×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
125	C₂³×C₆₇	#137	1	2	yes	—
126	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₆₇	—	2	2	no	—
128	C₂×C₄×C₆₇	—	2	1	no	—
129	C₂×C₄×C₆₇	—	2	1	no	—
130	C₂×C₄×C₆₇	—	2	1	no	—
131	C₂×C₄×C₆₇	—	2	1	no	—
132	C₂×C₄×C₆₇	—	2	2	no	—
133	C₂×C₄×C₆₇	—	2	2	no	—
134	C₂×C₄×C₆₇	—	2	8	no	—
135	C₂×C₄×C₆₇	—	2	4	no	—
136	C₂×C₄×C₆₇	—	2	4	no	—
137	C₂×C₄×C₆₇	—	2	4	no	—
138	C₂×C₄×C₆₇	—	2	4	no	—
139	C₂×C₄×C₆₇	—	4	4	no	—
140	C₂×C₄×C₆₇	—	4	8	no	—
141	C₂×C₄×C₆₇	—	4	2	no	—
142	C₂×C₄×C₆₇	—	4	2	no	—
143	C₂×C₄×C₆₇	—	4	4	no	—
144	C₂×C₄×C₆₇	—	4	8	no	—
145	C₂×C₄×C₆₇	—	4	4	no	—
146	C₂×C₄×C₆₇	—	4	4	no	—
147	C₂×C₄×C₆₇	—	4	4	no	—
148	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
149	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
150	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
151	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
152	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
153	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
154	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₄×C₆₇	2₋¹⁺⁴	1	4	yes	—
156	C₈×C₆₇	—	4	4	no	—
157	C₂²×C₆₇	—	2	2	no	—
158	C₂²×C₆₇	—	2	2	no	—
159	C₂²×C₆₇	—	2	4	no	—
160	C₂²×C₆₇	—	2	4	no	—
161	C₂²×C₆₇	#417	1	4	yes	—
162	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
163	C₂²×C₆₇	#536	1	2	yes	—
164	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
165	C₂²×C₆₇	C₂×#137	1	1	yes	characteristic
166	C₂²×C₆₇	#575	1	2	yes	—
167	C₄×C₆₇	—	2	2	no	—
168	C₄×C₆₇	—	2	2	no	—
169	C₄×C₆₇	—	2	2	no	—
170	C₄×C₆₇	—	2	2	no	—
171	C₄×C₆₇	—	2	2	no	—
172	C₄×C₆₇	—	4	4	no	—
173	C₄×C₆₇	—	4	4	no	—
174	C₄×C₆₇	—	8	4	no	—
175	C₄×C₆₇	#519	1	2	yes	—
176	C₄×C₆₇	#536	1	2	yes	—
177	#21838	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
178	C₂×C₆₇	—	2	4	no	—
179	C₂×C₆₇	#2310	1	1	yes	characteristic
180	C₂×C₆₇	C₂×#417	1	1	yes	characteristic
181	C₂×C₆₇	#2896	1	1	yes	characteristic
182	C₂×C₆₇	#3234	1	2	yes	—
183	C₂×C₆₇	#3236	1	2	yes	—
184	C₂²×#125	C₂×C₆₇	1	1	yes	characteristic
185	C₂×#432	C₂×C₆₇	1	1	yes	characteristic
186	#2285	C₂×C₆₇	1	1	yes	characteristic
187	#1992	C₂×C₆₇	1	4	yes	complemented
188	#1989	C₂×C₆₇	1	4	yes	complemented
189	#1824	C₂×C₆₇	1	2	yes	—
190	#1823	C₂×C₆₇	1	2	yes	—
191	C₆₇	#21838	1	1	yes	characteristic, direct product factor, Sylow
192	C₂×#125	—	2	1	no	—
193	C₂×#125	—	2	1	no	—
194	C₂×#125	—	2	4	no	—
195	C₂×#125	C₄×C₆₇	1	8	yes	complemented
196	C₂×#125	C₂²×C₆₇	1	2	yes	—
197	C₂×#125	C₂²×C₆₇	1	2	yes	—
198	C₂×#117	—	2	1	no	—
199	C₂²×C₄⋊C₄	—	2	1	no	—
200	C₂²×C₄⋊C₄	—	2	1	no	—
201	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
202	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
203	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
204	#432	—	2	4	no	—
205	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
206	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
207	C₂×#97	C₂²×C₆₇	1	2	yes	—
208	#384	C₂²×C₆₇	1	2	yes	—
209	#382	—	2	1	no	—
210	#380	—	2	1	no	—
211	#380	C₂²×C₆₇	1	2	yes	—
212	C₄×C₄⋊C₄	—	2	1	no	—
213	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
214	#340	C₂²×C₆₇	1	8	yes	—
215	#340	C₂²×C₆₇	1	8	yes	—
216	C₂³×C₄	—	2	1	no	—
217	C₂³×C₄	—	2	1	no	—
218	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
219	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
220	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
221	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
222	#125	—	2	8	no	—
223	#125	—	2	8	no	—
224	#125	—	2	4	no	—
225	#125	—	2	4	no	—
226	#125	C₂×C₄×C₆₇	1	8	yes	—
227	#125	C₂×C₄×C₆₇	1	8	yes	—
228	#117	—	4	4	no	—
229	C₂×C₄⋊C₄	—	2	8	no	—
230	C₂×C₄⋊C₄	—	2	8	no	—
231	C₂×C₄⋊C₄	—	2	2	no	—
232	C₂×C₄⋊C₄	—	2	2	no	—
233	C₂×C₄⋊C₄	—	2	2	no	—
234	C₂×C₄⋊C₄	—	2	2	no	—
235	C₂×C₄⋊C₄	—	2	1	no	—
236	C₂×C₄⋊C₄	—	2	1	no	—
237	C₂×C₄⋊C₄	—	2	2	no	—
238	C₂×C₄⋊C₄	—	2	2	no	—
239	C₂×C₄⋊C₄	—	2	2	no	—
240	C₂×C₄⋊C₄	—	2	2	no	—
241	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
242	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
243	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
244	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
245	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
246	C₂×C₂²⋊C₄	—	2	2	no	—
247	C₂×C₄²	—	2	2	no	—
248	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
249	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
250	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
251	C₂×C₄²	D₄×C₆₇	1	4	yes	—
252	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
253	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
254	C₄⋊C₈	—	2	4	no	—
255	C₂²⋊C₈	—	2	4	no	—
256	#97	—	2	4	no	—
257	#95	—	2	2	no	—
258	#95	—	2	2	no	—
259	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
260	C₂²×C₄	—	2	2	no	—
261	C₂²×C₄	—	2	2	no	—
262	C₂²×C₄	—	2	4	no	—
263	C₂²×C₄	—	2	4	no	—
264	C₂²×C₄	—	2	4	no	—
265	C₂²×C₄	—	2	2	no	—
266	C₂²×C₄	—	2	2	no	—
267	C₂²×C₄	—	2	4	no	—
268	C₂²×C₄	—	4	2	no	—
269	C₂²×C₄	—	4	4	no	—
270	C₂²×C₄	—	4	2	no	—
271	C₂²×C₄	—	4	2	no	—
272	C₂²×C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
273	C₂²×C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
274	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
275	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
276	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
277	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
278	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
279	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
280	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
281	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
282	C₂×C₈	—	2	4	no	—
283	C₂×C₈	—	2	8	no	—
284	C₄⋊C₄	—	2	8	no	—
285	C₄⋊C₄	—	2	8	no	—
286	C₄⋊C₄	—	2	8	no	—
287	C₄⋊C₄	—	2	8	no	—
288	C₄⋊C₄	—	2	4	no	—
289	C₄⋊C₄	—	2	4	no	—
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	2	8	no	—
293	C₄⋊C₄	—	4	4	no	—
294	C₄⋊C₄	—	4	4	no	—
295	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
296	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
297	C₂²⋊C₄	—	4	4	no	—
298	C₄²	—	4	4	no	—
299	C₄²	C₂²⋊C₄×C₆₇	1	4	yes	—
300	C₄²	C₂²⋊C₄×C₆₇	1	4	yes	—
301	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
302	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
303	C₄²	C₂×D₄×C₆₇	1	2	yes	—
304	C₄²	C₂×D₄×C₆₇	1	2	yes	—
305	C₂³	—	2	2	no	—
306	C₂³	—	2	2	no	—
307	C₂³	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
308	C₂³	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
309	C₂³	#137×C₆₇	1	2	yes	—
310	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	8	no	—
319	C₂×C₄	—	2	4	no	—
320	C₂×C₄	—	2	4	no	—
321	C₂×C₄	—	2	4	no	—
322	C₂×C₄	—	2	4	no	—
323	C₂×C₄	—	4	4	no	—
324	C₂×C₄	—	4	8	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	4	4	no	—
328	C₂×C₄	—	4	8	no	—
329	C₂×C₄	—	4	4	no	—
330	C₂×C₄	—	4	4	no	—
331	C₂×C₄	—	4	4	no	—
332	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
333	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
334	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
335	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
336	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
337	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
338	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
339	C₂×C₄	2₋¹⁺⁴×C₆₇	1	4	yes	—
340	C₈	—	4	4	no	—
341	C₂²	—	2	2	no	—
342	C₂²	—	2	2	no	—
343	C₂²	—	2	4	no	—
344	C₂²	—	2	4	no	—
345	C₂²	#417×C₆₇	1	4	yes	—
346	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
347	C₂²	#536×C₆₇	1	2	yes	—
348	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
349	C₂²	C₂×#137×C₆₇	1	1	yes	characteristic
350	C₂²	#575×C₆₇	1	2	yes	—
351	C₄	—	2	2	no	—
352	C₄	—	2	2	no	—
353	C₄	—	2	2	no	—
354	C₄	—	2	2	no	—
355	C₄	—	2	2	no	—
356	C₄	—	4	4	no	—
357	C₄	—	4	4	no	—
358	C₄	—	8	4	no	—
359	C₄	#519×C₆₇	1	2	yes	—
360	C₄	#536×C₆₇	1	2	yes	—
361	C₂	—	2	4	no	—
362	C₂	C₆₇×#2310	1	1	yes	characteristic
363	C₂	C₂×#417×C₆₇	1	1	yes	characteristic
364	C₂	C₆₇×#2896	1	1	yes	characteristic
365	C₂	C₆₇×#3234	1	2	yes	—
366	C₂	C₆₇×#3236	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×#1992	C₂	semidirect
5. C₆₇×#1989	C₂	semidirect
11. C₂×#125×C₆₇	C₄	semidirect
177. #21838	C₆₇	direct (N × H)
187. #1992	C₂×C₆₇	semidirect
188. #1989	C₂×C₆₇	semidirect
191. C₆₇	#21838	direct (N × H)
195. C₂×#125	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×#125×C₆₇	2
C₂×#432×C₆₇	2
C₆₇×#2285	2
C₆₇×#1992	2
C₆₇×#1989	2
C₆₇×#1824	2
C₆₇×#1823	2
#21838	67

Subgroups — C67×#21838 #30701089

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21838 #30701089