Subgroups — C₈×#117×C₆₇ #30699722

Order 17152.

1174 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#431×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₄×#117×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₈×C₄⋊C₄×C₆₇	C₂	1	4	yes	maximal, complemented
5	C₈×C₂²⋊C₄×C₆₇	C₂	1	4	yes	maximal
6	C₆₇×#1744	C₂	1	2	yes	maximal, complemented
7	C₄²×C₈×C₆₇	C₂	1	2	yes	maximal, complemented
8	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
9	C₂×#117×C₆₇	C₄	1	2	yes	—
10	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
11	#431×C₆₇	C₄	1	8	yes	complemented
12	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
13	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₆₇	C₄	1	8	yes	complemented
15	C₂×C₄×C₈×C₆₇	C₂²	1	8	yes	—
16	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₄×C₄⋊C₄×C₆₇	C₄	1	2	yes	—
21	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
22	C₄×C₂²⋊C₄×C₆₇	C₄	1	2	yes	—
23	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
24	#376×C₆₇	C₄	1	2	yes	—
25	C₄³×C₆₇	C₂²	1	2	yes	—
26	#336×C₆₇	C₂²	1	8	yes	—
27	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
30	C₂²×C₈×C₆₇	—	2	8	no	—
31	C₂²×C₈×C₆₇	C₂×C₄	1	8	yes	—
32	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
33	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
34	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
35	#117×C₆₇	C₈	1	16	yes	direct product factor
36	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
37	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
38	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
39	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
42	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
43	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
44	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
45	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
47	C₄⋊C₈×C₆₇	C₂×C₄	1	8	yes	—
48	C₂²⋊C₈×C₆₇	C₂×C₄	1	8	yes	—
49	C₄×C₈×C₆₇	—	2	16	no	—
50	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
51	C₄×C₈×C₆₇	C₂×C₄	1	8	yes	—
52	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
53	C₄×C₈×C₆₇	C₂×C₄	1	8	yes	—
54	#95×C₆₇	C₂×C₄	1	4	yes	—
55	#95×C₆₇	C₂×C₄	1	4	yes	—
56	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₆₇	—	2	4	no	—
58	C₂²×C₄×C₆₇	—	2	4	no	—
59	C₂²×C₄×C₆₇	C₄²	1	4	yes	—
60	C₂²×C₄×C₆₇	C₄²	1	4	yes	—
61	C₂²×C₄×C₆₇	C₂×C₈	1	8	yes	—
62	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
67	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
71	C₂×C₈×C₆₇	—	2	4	no	—
72	C₂×C₈×C₆₇	—	2	8	no	—
73	C₂×C₈×C₆₇	—	2	16	no	—
74	C₂×C₈×C₆₇	C₂²×C₄	1	8	yes	—
75	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
76	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
77	C₂×C₈×C₆₇	#41	1	8	yes	—
78	C₄⋊C₄×C₆₇	C₂×C₈	1	16	yes	complemented
79	C₂²⋊C₄×C₆₇	C₂×C₈	1	16	yes	—
80	C₄²×C₆₇	—	2	8	no	—
81	C₄²×C₆₇	—	2	8	no	—
82	C₄²×C₆₇	C₄²	1	4	yes	—
83	C₄²×C₆₇	C₄²	1	4	yes	—
84	C₄²×C₆₇	C₂×C₈	1	16	yes	complemented
85	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
86	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
87	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
88	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
89	C₂³×C₆₇	—	2	2	no	—
90	C₂³×C₆₇	—	2	2	no	—
91	C₂³×C₆₇	C₂×C₄²	1	2	yes	—
92	C₂³×C₆₇	C₂²×C₈	1	4	yes	—
93	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
94	C₂×C₄×C₆₇	—	2	8	no	—
95	C₂×C₄×C₆₇	—	2	2	no	—
96	C₂×C₄×C₆₇	—	2	8	no	—
97	C₂×C₄×C₆₇	—	2	2	no	—
98	C₂×C₄×C₆₇	—	2	4	no	—
99	C₂×C₄×C₆₇	—	2	4	no	—
100	C₂×C₄×C₆₇	—	2	8	no	—
101	C₂×C₄×C₆₇	C₄×C₈	1	16	yes	complemented
102	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
103	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
104	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
105	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
106	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
107	C₂×C₄×C₆₇	C₂²×C₈	1	16	yes	—
108	C₂×C₄×C₆₇	C₂²×C₈	1	4	yes	—
109	C₂×C₄×C₆₇	C₂²×C₈	1	4	yes	—
110	C₂×C₄×C₆₇	#131	1	4	yes	—
111	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
113	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
114	C₈×C₆₇	—	2	4	no	—
115	C₈×C₆₇	—	2	8	no	—
116	C₈×C₆₇	#117	1	8	yes	direct product factor
117	C₂²×C₆₇	—	2	2	no	—
118	C₂²×C₆₇	—	2	8	no	—
119	C₂²×C₆₇	—	2	2	no	—
120	C₂²×C₆₇	C₂×C₄×C₈	1	4	yes	—
121	C₂²×C₆₇	C₂²×C₄²	1	1	yes	characteristic
122	C₂²×C₆₇	C₄×#41	1	2	yes	—
123	C₂²×C₆₇	C₂³×C₈	1	2	yes	—
124	C₂²×C₆₇	C₂×#131	1	2	yes	—
125	C₄×C₆₇	—	2	2	no	—
126	C₄×C₆₇	—	2	16	no	—
127	C₄×C₆₇	—	2	2	no	—
128	C₄×C₆₇	C₂×C₄×C₈	1	4	yes	—
129	C₄×C₆₇	C₂×C₄×C₈	1	4	yes	—
130	C₄×C₆₇	#405	1	4	yes	—
131	C₄×C₆₇	C₂×#117	1	2	yes	—
132	C₄×C₆₇	C₄×#41	1	2	yes	—
133	C₈×#117	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
134	C₂×C₆₇	—	2	4	no	—
135	C₂×C₆₇	C₄×#117	1	1	yes	characteristic
136	C₂×C₆₇	C₂²×C₄×C₈	1	1	yes	characteristic
137	C₂×C₆₇	C₂×#405	1	1	yes	characteristic
138	C₂×C₆₇	C₈×#41	1	4	yes	—
139	C₂×#431	C₂×C₆₇	1	1	yes	characteristic
140	C₂²×C₄×C₈	C₂×C₆₇	1	1	yes	characteristic
141	C₄×#117	C₂×C₆₇	1	1	yes	characteristic
142	C₈×C₄⋊C₄	C₂×C₆₇	1	4	yes	complemented
143	C₈×C₂²⋊C₄	C₂×C₆₇	1	4	yes	—
144	#1744	C₂×C₆₇	1	2	yes	complemented
145	C₄²×C₈	C₂×C₆₇	1	2	yes	complemented
146	C₆₇	C₈×#117	1	1	yes	characteristic, direct product factor, Sylow
147	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
148	C₂×#117	C₄×C₆₇	1	2	yes	—
149	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
150	#431	C₄×C₆₇	1	8	yes	complemented
151	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
152	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
153	C₂×C₄×C₈	C₄×C₆₇	1	8	yes	complemented
154	C₂×C₄×C₈	C₂²×C₆₇	1	8	yes	—
155	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
156	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
157	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
158	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
159	C₄×C₄⋊C₄	C₄×C₆₇	1	2	yes	—
160	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
161	C₄×C₂²⋊C₄	C₄×C₆₇	1	2	yes	—
162	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
163	#376	C₄×C₆₇	1	2	yes	—
164	C₄³	C₂²×C₆₇	1	2	yes	—
165	#336	C₂²×C₆₇	1	8	yes	—
166	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
167	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
168	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
169	C₂²×C₈	—	2	8	no	—
170	C₂²×C₈	C₂×C₄×C₆₇	1	8	yes	—
171	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
172	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
173	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
174	#117	C₈×C₆₇	1	16	yes	direct product factor
175	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
176	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
177	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
178	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
179	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
180	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
181	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
182	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
183	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
184	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
185	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
186	C₄⋊C₈	C₂×C₄×C₆₇	1	8	yes	—
187	C₂²⋊C₈	C₂×C₄×C₆₇	1	8	yes	—
188	C₄×C₈	—	2	16	no	—
189	C₄×C₈	C₂×C₄×C₆₇	1	4	yes	—
190	C₄×C₈	C₂×C₄×C₆₇	1	8	yes	—
191	C₄×C₈	C₂×C₄×C₆₇	1	4	yes	—
192	C₄×C₈	C₂×C₄×C₆₇	1	8	yes	—
193	#95	C₂×C₄×C₆₇	1	4	yes	—
194	#95	C₂×C₄×C₆₇	1	4	yes	—
195	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
196	C₂²×C₄	—	2	4	no	—
197	C₂²×C₄	—	2	4	no	—
198	C₂²×C₄	C₄²×C₆₇	1	4	yes	—
199	C₂²×C₄	C₄²×C₆₇	1	4	yes	—
200	C₂²×C₄	C₂×C₈×C₆₇	1	8	yes	—
201	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
202	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
203	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
204	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
205	C₂²×C₄	C₂²×C₄×C₆₇	1	8	yes	—
206	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
207	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
208	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
209	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
210	C₂×C₈	—	2	8	no	—
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	—	2	16	no	—
213	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
214	C₂×C₈	C₂²×C₄×C₆₇	1	8	yes	—
215	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
216	C₂×C₈	#41×C₆₇	1	8	yes	—
217	C₄⋊C₄	C₂×C₈×C₆₇	1	16	yes	complemented
218	C₂²⋊C₄	C₂×C₈×C₆₇	1	16	yes	—
219	C₄²	—	2	8	no	—
220	C₄²	—	2	8	no	—
221	C₄²	C₄²×C₆₇	1	4	yes	—
222	C₄²	C₄²×C₆₇	1	4	yes	—
223	C₄²	C₂×C₈×C₆₇	1	16	yes	complemented
224	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
225	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
226	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
227	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
228	C₂³	—	2	2	no	—
229	C₂³	—	2	2	no	—
230	C₂³	C₂×C₄²×C₆₇	1	2	yes	—
231	C₂³	C₂²×C₈×C₆₇	1	4	yes	—
232	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
233	C₂×C₄	—	2	8	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	8	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	8	no	—
240	C₂×C₄	C₄×C₈×C₆₇	1	16	yes	complemented
241	C₂×C₄	C₂×C₄²×C₆₇	1	2	yes	—
242	C₂×C₄	C₂×C₄²×C₆₇	1	2	yes	—
243	C₂×C₄	C₂×C₄²×C₆₇	1	2	yes	—
244	C₂×C₄	C₂×C₄²×C₆₇	1	2	yes	—
245	C₂×C₄	C₂×C₄²×C₆₇	1	2	yes	—
246	C₂×C₄	C₂²×C₈×C₆₇	1	16	yes	—
247	C₂×C₄	C₂²×C₈×C₆₇	1	4	yes	—
248	C₂×C₄	C₂²×C₈×C₆₇	1	4	yes	—
249	C₂×C₄	#131×C₆₇	1	4	yes	—
250	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
251	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
252	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
253	C₈	—	2	4	no	—
254	C₈	—	2	8	no	—
255	C₈	#117×C₆₇	1	8	yes	direct product factor
256	C₂²	—	2	2	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	2	8	no	—
259	C₂²	C₂×C₄×C₈×C₆₇	1	4	yes	—
260	C₂²	C₂²×C₄²×C₆₇	1	1	yes	characteristic
261	C₂²	C₄×#41×C₆₇	1	2	yes	—
262	C₂²	C₂³×C₈×C₆₇	1	2	yes	—
263	C₂²	C₂×#131×C₆₇	1	2	yes	—
264	C₄	—	2	2	no	—
265	C₄	—	2	2	no	—
266	C₄	—	2	16	no	—
267	C₄	C₂×C₄×C₈×C₆₇	1	4	yes	—
268	C₄	C₂×C₄×C₈×C₆₇	1	4	yes	—
269	C₄	#405×C₆₇	1	4	yes	—
270	C₄	C₂×#117×C₆₇	1	2	yes	—
271	C₄	C₄×#41×C₆₇	1	2	yes	—
272	C₂	—	2	4	no	—
273	C₂	C₄×#117×C₆₇	1	1	yes	characteristic
274	C₂	C₂²×C₄×C₈×C₆₇	1	1	yes	characteristic
275	C₂	C₂×#405×C₆₇	1	1	yes	characteristic
276	C₂	C₈×#41×C₆₇	1	4	yes	—

	subgroup	G/H
center	C₂×C₄×C₈×C₆₇	C₂²
commutator	C₂	C₂²×C₄×C₈×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂³×C₆₇	C₂³×C₄
prime radical	C₂⁴×C₆₇	C₂²×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₈×C₄⋊C₄×C₆₇	C₂	semidirect
6. C₆₇×#1744	C₂	semidirect
7. C₄²×C₈×C₆₇	C₂	semidirect
11. #431×C₆₇	C₄	semidirect
14. C₂×C₄×C₈×C₆₇	C₄	semidirect
35. #117×C₆₇	C₈	direct (N × H)
78. C₄⋊C₄×C₆₇	C₂×C₈	semidirect
84. C₄²×C₆₇	C₂×C₈	semidirect
101. C₂×C₄×C₆₇	C₄×C₈	semidirect
116. C₈×C₆₇	#117	direct (N × H)
133. C₈×#117	C₆₇	direct (N × H)
142. C₈×C₄⋊C₄	C₂×C₆₇	semidirect
144. #1744	C₂×C₆₇	semidirect
145. C₄²×C₈	C₂×C₆₇	semidirect
146. C₆₇	C₈×#117	direct (N × H)
150. #431	C₄×C₆₇	semidirect
153. C₂×C₄×C₈	C₄×C₆₇	semidirect
174. #117	C₈×C₆₇	direct (N × H)
217. C₄⋊C₄	C₂×C₈×C₆₇	semidirect
223. C₄²	C₂×C₈×C₆₇	semidirect
240. C₂×C₄	C₄×C₈×C₆₇	semidirect
255. C₈	#117×C₆₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#431×C₆₇	2
C₂²×C₄×C₈×C₆₇	2
C₄×#117×C₆₇	2
C₈×C₄⋊C₄×C₆₇	2
C₈×C₂²⋊C₄×C₆₇	2
C₆₇×#1744	2
C₄²×C₈×C₆₇	2
C₈×#117	67

Subgroups — C8×#117×C67 #30699722

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×#117×C₆₇ #30699722