Subgroups — C₂×C₄×C₄⋊C₈×C₆₇ #30699674

Order 17152.

1550 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊C₈×C₆₇	C₂	1	2	yes	maximal
2	C₂²×C₄×C₈×C₆₇	C₂	1	2	yes	maximal
3	C₂×C₄³×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₄×C₄⋊C₈×C₆₇	C₂	1	8	yes	maximal, direct product factor
5	C₂×#336×C₆₇	C₂	1	2	yes	maximal
6	C₂³×C₈×C₆₇	C₂²	1	4	yes	—
7	C₂²×C₄²×C₆₇	C₄	1	2	yes	—
8	C₂²×C₄²×C₆₇	C₄	1	2	yes	—
9	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
10	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
11	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₈×C₆₇	C₄	1	16	yes	direct product factor
13	C₂×C₄⋊C₈×C₆₇	C₂²	1	8	yes	—
14	C₂×C₄×C₈×C₆₇	—	2	8	no	—
15	C₂×C₄×C₈×C₆₇	C₂²	1	8	yes	—
16	C₄³×C₆₇	C₄	1	4	yes	—
17	C₄³×C₆₇	C₂²	1	4	yes	—
18	#336×C₆₇	C₂²	1	8	yes	—
19	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
20	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
21	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
23	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
24	C₂²×C₈×C₆₇	—	2	8	no	—
25	C₂²×C₈×C₆₇	—	2	8	no	—
26	C₂²×C₈×C₆₇	C₂×C₄	1	16	yes	—
27	C₂²×C₈×C₆₇	C₂³	1	8	yes	—
28	C₂×C₄²×C₆₇	—	2	8	no	—
29	C₂×C₄²×C₆₇	—	2	4	no	—
30	C₂×C₄²×C₆₇	C₈	1	16	yes	complemented
31	C₂×C₄²×C₆₇	C₂×C₄	1	8	yes	—
32	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₆₇	C₂×C₄	1	8	yes	—
35	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₆₇	C₂×C₄	1	8	yes	—
37	C₂×C₄²×C₆₇	D₄	1	4	yes	—
38	C₂×C₄²×C₆₇	Q₈	1	4	yes	—
39	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
40	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
41	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
42	C₄⋊C₈×C₆₇	C₂×C₄	1	32	yes	direct product factor
43	C₄×C₈×C₆₇	—	2	16	no	—
44	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₆₇	—	2	4	no	—
46	C₂²×C₄×C₆₇	—	2	4	no	—
47	C₂²×C₄×C₆₇	—	2	4	no	—
48	C₂²×C₄×C₆₇	C₄²	1	4	yes	—
49	C₂²×C₄×C₆₇	C₄²	1	4	yes	—
50	C₂²×C₄×C₆₇	C₄⋊C₄	1	8	yes	—
51	C₂²×C₄×C₆₇	C₂×C₈	1	8	yes	—
52	C₂²×C₄×C₆₇	C₂×C₈	1	8	yes	—
53	C₂²×C₄×C₆₇	C₂×C₈	1	8	yes	—
54	C₂²×C₄×C₆₇	M₁₆	1	8	yes	—
55	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
56	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
57	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
60	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
61	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
63	C₂²×C₄×C₆₇	#41	1	4	yes	—
64	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
65	C₂×C₈×C₆₇	—	2	32	no	—
66	C₂×C₈×C₆₇	—	2	8	no	—
67	C₂×C₈×C₆₇	—	2	8	no	—
68	C₂×C₈×C₆₇	C₂²×C₄	1	16	yes	—
69	C₄²×C₆₇	—	2	4	no	—
70	C₄²×C₆₇	—	2	16	no	—
71	C₄²×C₆₇	—	2	4	no	—
72	C₄²×C₆₇	C₄²	1	8	yes	—
73	C₄²×C₆₇	C₄⋊C₄	1	8	yes	—
74	C₄²×C₆₇	C₂×C₈	1	32	yes	complemented
75	C₄²×C₆₇	C₂²×C₄	1	8	yes	—
76	C₄²×C₆₇	C₂×D₄	1	4	yes	—
77	C₄²×C₆₇	C₂×Q₈	1	4	yes	—
78	C₂³×C₆₇	C₂×C₄²	1	2	yes	—
79	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
80	C₂³×C₆₇	#117	1	2	yes	—
81	C₂³×C₆₇	C₂²×C₈	1	4	yes	—
82	C₂³×C₆₇	C₂×M₁₆	1	4	yes	—
83	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
84	C₂×C₄×C₆₇	—	2	4	no	—
85	C₂×C₄×C₆₇	—	2	4	no	—
86	C₂×C₄×C₆₇	—	2	4	no	—
87	C₂×C₄×C₆₇	—	2	8	no	—
88	C₂×C₄×C₆₇	—	2	16	no	—
89	C₂×C₄×C₆₇	C₄×C₈	1	16	yes	complemented
90	C₂×C₄×C₆₇	C₄⋊C₈	1	32	yes	direct product factor
91	C₂×C₄×C₆₇	C₂×C₄²	1	4	yes	—
92	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
93	C₂×C₄×C₆₇	C₂×C₄²	1	4	yes	—
94	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
95	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	8	yes	—
96	C₂×C₄×C₆₇	#117	1	2	yes	—
97	C₂×C₄×C₆₇	C₄×D₄	1	8	yes	—
98	C₂×C₄×C₆₇	C₄×Q₈	1	8	yes	—
99	C₂×C₄×C₆₇	C₂²×C₈	1	8	yes	—
100	C₂×C₄×C₆₇	C₂²×C₈	1	8	yes	—
101	C₂×C₄×C₆₇	C₂²×C₈	1	8	yes	—
102	C₂×C₄×C₆₇	C₂×M₁₆	1	8	yes	—
103	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
104	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
105	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
106	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
107	C₈×C₆₇	—	2	16	no	—
108	C₂²×C₆₇	C₄×C₄⋊C₄	1	4	yes	—
109	C₂²×C₆₇	C₂×C₄×C₈	1	4	yes	—
110	C₂²×C₆₇	C₄×M₁₆	1	4	yes	—
111	C₂²×C₆₇	C₂×C₄⋊C₈	1	8	yes	—
112	C₂²×C₆₇	#431	1	8	yes	—
113	C₂²×C₆₇	C₂²×C₄²	1	1	yes	characteristic
114	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
115	C₂²×C₆₇	C₂×#117	1	1	yes	characteristic
116	C₂²×C₆₇	C₂³×C₈	1	2	yes	—
117	C₂²×C₆₇	C₂²×M₁₆	1	2	yes	—
118	C₄×C₆₇	—	2	4	no	—
119	C₄×C₆₇	—	2	8	no	—
120	C₄×C₆₇	C₄×C₄⋊C₄	1	4	yes	—
121	C₄×C₆₇	C₂×C₄×C₈	1	8	yes	complemented
122	C₄×C₆₇	C₂×C₄⋊C₈	1	16	yes	direct product factor
123	C₄×C₆₇	C₂×C₄×D₄	1	2	yes	—
124	C₄×C₆₇	C₂×C₄×Q₈	1	2	yes	—
125	C₂×C₄×C₄⋊C₈	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
126	C₂×C₆₇	C₄×C₄⋊C₈	1	8	yes	direct product factor
127	C₂×C₆₇	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₆₇	C₂²×C₄×C₈	1	1	yes	characteristic
129	C₂×C₆₇	C₂×C₄×M₁₆	1	1	yes	characteristic
130	C₂×C₆₇	C₂²×C₄⋊C₈	1	2	yes	—
131	C₂×C₆₇	C₂×#431	1	2	yes	—
132	C₂²×C₄⋊C₈	C₂×C₆₇	1	2	yes	—
133	C₂²×C₄×C₈	C₂×C₆₇	1	2	yes	—
134	C₂×C₄³	C₂×C₆₇	1	1	yes	characteristic
135	C₄×C₄⋊C₈	C₂×C₆₇	1	8	yes	direct product factor
136	C₂×#336	C₂×C₆₇	1	2	yes	—
137	C₆₇	C₂×C₄×C₄⋊C₈	1	1	yes	characteristic, direct product factor, Sylow
138	C₂³×C₈	C₂²×C₆₇	1	4	yes	—
139	C₂²×C₄²	C₄×C₆₇	1	2	yes	—
140	C₂²×C₄²	C₄×C₆₇	1	2	yes	—
141	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
142	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
143	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
144	C₂×C₄⋊C₈	C₄×C₆₇	1	16	yes	direct product factor
145	C₂×C₄⋊C₈	C₂²×C₆₇	1	8	yes	—
146	C₂×C₄×C₈	—	2	8	no	—
147	C₂×C₄×C₈	C₂²×C₆₇	1	8	yes	—
148	C₄³	C₄×C₆₇	1	4	yes	—
149	C₄³	C₂²×C₆₇	1	4	yes	—
150	#336	C₂²×C₆₇	1	8	yes	—
151	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
152	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
153	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
154	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
155	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
156	C₂²×C₈	—	2	8	no	—
157	C₂²×C₈	—	2	8	no	—
158	C₂²×C₈	C₂×C₄×C₆₇	1	16	yes	—
159	C₂²×C₈	C₂³×C₆₇	1	8	yes	—
160	C₂×C₄²	—	2	8	no	—
161	C₂×C₄²	—	2	4	no	—
162	C₂×C₄²	C₈×C₆₇	1	16	yes	complemented
163	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
164	C₂×C₄²	C₂×C₄×C₆₇	1	8	yes	—
165	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
166	C₂×C₄²	C₂×C₄×C₆₇	1	8	yes	—
167	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
168	C₂×C₄²	C₂×C₄×C₆₇	1	8	yes	—
169	C₂×C₄²	D₄×C₆₇	1	4	yes	—
170	C₂×C₄²	Q₈×C₆₇	1	4	yes	—
171	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
172	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
173	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
174	C₄⋊C₈	C₂×C₄×C₆₇	1	32	yes	direct product factor
175	C₄×C₈	—	2	16	no	—
176	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
177	C₂²×C₄	—	2	4	no	—
178	C₂²×C₄	—	2	4	no	—
179	C₂²×C₄	—	2	4	no	—
180	C₂²×C₄	C₄²×C₆₇	1	4	yes	—
181	C₂²×C₄	C₄²×C₆₇	1	4	yes	—
182	C₂²×C₄	C₄⋊C₄×C₆₇	1	8	yes	—
183	C₂²×C₄	C₂×C₈×C₆₇	1	8	yes	—
184	C₂²×C₄	C₂×C₈×C₆₇	1	8	yes	—
185	C₂²×C₄	C₂×C₈×C₆₇	1	8	yes	—
186	C₂²×C₄	M₁₆×C₆₇	1	8	yes	—
187	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
188	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
189	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
190	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
191	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
192	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
193	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
194	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
195	C₂²×C₄	#41×C₆₇	1	4	yes	—
196	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
197	C₂×C₈	—	2	32	no	—
198	C₂×C₈	—	2	8	no	—
199	C₂×C₈	—	2	8	no	—
200	C₂×C₈	C₂²×C₄×C₆₇	1	16	yes	—
201	C₄²	—	2	4	no	—
202	C₄²	—	2	16	no	—
203	C₄²	—	2	4	no	—
204	C₄²	C₄²×C₆₇	1	8	yes	—
205	C₄²	C₄⋊C₄×C₆₇	1	8	yes	—
206	C₄²	C₂×C₈×C₆₇	1	32	yes	complemented
207	C₄²	C₂²×C₄×C₆₇	1	8	yes	—
208	C₄²	C₂×D₄×C₆₇	1	4	yes	—
209	C₄²	C₂×Q₈×C₆₇	1	4	yes	—
210	C₂³	C₂×C₄²×C₆₇	1	2	yes	—
211	C₂³	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
212	C₂³	#117×C₆₇	1	2	yes	—
213	C₂³	C₂²×C₈×C₆₇	1	4	yes	—
214	C₂³	C₂×M₁₆×C₆₇	1	4	yes	—
215	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	—	2	4	no	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	2	8	no	—
220	C₂×C₄	—	2	16	no	—
221	C₂×C₄	C₄×C₈×C₆₇	1	16	yes	complemented
222	C₂×C₄	C₄⋊C₈×C₆₇	1	32	yes	direct product factor
223	C₂×C₄	C₂×C₄²×C₆₇	1	4	yes	—
224	C₂×C₄	C₂×C₄²×C₆₇	1	2	yes	—
225	C₂×C₄	C₂×C₄²×C₆₇	1	4	yes	—
226	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
227	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	8	yes	—
228	C₂×C₄	#117×C₆₇	1	2	yes	—
229	C₂×C₄	C₄×D₄×C₆₇	1	8	yes	—
230	C₂×C₄	C₄×Q₈×C₆₇	1	8	yes	—
231	C₂×C₄	C₂²×C₈×C₆₇	1	8	yes	—
232	C₂×C₄	C₂²×C₈×C₆₇	1	8	yes	—
233	C₂×C₄	C₂²×C₈×C₆₇	1	8	yes	—
234	C₂×C₄	C₂×M₁₆×C₆₇	1	8	yes	—
235	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
236	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
237	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
238	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
239	C₈	—	2	16	no	—
240	C₂²	C₄×C₄⋊C₄×C₆₇	1	4	yes	—
241	C₂²	C₂×C₄×C₈×C₆₇	1	4	yes	—
242	C₂²	C₄×M₁₆×C₆₇	1	4	yes	—
243	C₂²	C₂×C₄⋊C₈×C₆₇	1	8	yes	—
244	C₂²	#431×C₆₇	1	8	yes	—
245	C₂²	C₂²×C₄²×C₆₇	1	1	yes	characteristic
246	C₂²	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
247	C₂²	C₂×#117×C₆₇	1	1	yes	characteristic
248	C₂²	C₂³×C₈×C₆₇	1	2	yes	—
249	C₂²	C₂²×M₁₆×C₆₇	1	2	yes	—
250	C₄	—	2	4	no	—
251	C₄	—	2	8	no	—
252	C₄	C₄×C₄⋊C₄×C₆₇	1	4	yes	—
253	C₄	C₂×C₄×C₈×C₆₇	1	8	yes	complemented
254	C₄	C₂×C₄⋊C₈×C₆₇	1	16	yes	direct product factor
255	C₄	C₂×C₄×D₄×C₆₇	1	2	yes	—
256	C₄	C₂×C₄×Q₈×C₆₇	1	2	yes	—
257	C₂	C₄×C₄⋊C₈×C₆₇	1	8	yes	direct product factor
258	C₂	C₂×C₄×C₄⋊C₄×C₆₇	1	1	yes	characteristic
259	C₂	C₂²×C₄×C₈×C₆₇	1	1	yes	characteristic
260	C₂	C₂×C₄×M₁₆×C₆₇	1	1	yes	characteristic
261	C₂	C₂²×C₄⋊C₈×C₆₇	1	2	yes	—
262	C₂	C₂×#431×C₆₇	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄²×C₆₇	C₂²
commutator	C₂	C₂²×C₄×C₈×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle, prime radical	C₂⁴×C₆₇	C₂²×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₄×C₄⋊C₈×C₆₇	C₂	direct (N × H)
12. C₂×C₄⋊C₈×C₆₇	C₄	direct (N × H)
30. C₂×C₄²×C₆₇	C₈	semidirect
42. C₄⋊C₈×C₆₇	C₂×C₄	direct (N × H)
74. C₄²×C₆₇	C₂×C₈	semidirect
89. C₂×C₄×C₆₇	C₄×C₈	semidirect
90. C₂×C₄×C₆₇	C₄⋊C₈	direct (N × H)
121. C₄×C₆₇	C₂×C₄×C₈	semidirect
122. C₄×C₆₇	C₂×C₄⋊C₈	direct (N × H)
125. C₂×C₄×C₄⋊C₈	C₆₇	direct (N × H)
126. C₂×C₆₇	C₄×C₄⋊C₈	direct (N × H)
135. C₄×C₄⋊C₈	C₂×C₆₇	direct (N × H)
137. C₆₇	C₂×C₄×C₄⋊C₈	direct (N × H)
144. C₂×C₄⋊C₈	C₄×C₆₇	direct (N × H)
162. C₂×C₄²	C₈×C₆₇	semidirect
174. C₄⋊C₈	C₂×C₄×C₆₇	direct (N × H)
206. C₄²	C₂×C₈×C₆₇	semidirect
221. C₂×C₄	C₄×C₈×C₆₇	semidirect
222. C₂×C₄	C₄⋊C₈×C₆₇	direct (N × H)
253. C₄	C₂×C₄×C₈×C₆₇	semidirect
254. C₄	C₂×C₄⋊C₈×C₆₇	direct (N × H)
257. C₂	C₄×C₄⋊C₈×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
C₂²×C₄⋊C₈×C₆₇	2
C₂²×C₄×C₈×C₆₇	2
C₂×C₄³×C₆₇	2
C₄×C₄⋊C₈×C₆₇	2
C₂×#336×C₆₇	2
C₂×C₄×C₄⋊C₈	67

Subgroups — C2×C4×C4⋊C8×C67 #30699674

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×C₄⋊C₈×C₆₇ #30699674