Subgroups — C₆₇×#14873 #30694124

Order 17152.

4290 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#393×C₆₇	C₂	1	4	yes	maximal, complemented
3	C₂×#386×C₆₇	C₂	1	3	yes	maximal
4	C₆₇×#1446	C₂	1	4	yes	maximal, complemented
5	C₆₇×C₂²⋊#95	C₂	1	3	yes	maximal, complemented
6	C₂³×Q₈×C₆₇	—	2	16	no	—
7	C₂³×Q₈×C₆₇	—	2	12	no	—
8	C₂³×Q₈×C₆₇	C₂²	1	4	yes	—
9	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
10	C₂⁴×C₄×C₆₇	C₂²	1	3	yes	—
11	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	6	yes	—
12	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	3	yes	—
13	#393×C₆₇	—	2	16	no	—
14	#386×C₆₇	—	2	12	no	—
15	C₂×#95×C₆₇	C₂²	1	6	yes	—
16	C₂×#95×C₆₇	C₂²	1	12	yes	—
17	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
18	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
19	C₂²×Q₈×C₆₇	—	2	192	no	—
20	C₂²×Q₈×C₆₇	—	2	32	no	—
21	C₂²×Q₈×C₆₇	—	2	24	no	—
22	C₂²×Q₈×C₆₇	—	2	24	no	—
23	C₂²×Q₈×C₆₇	D₄	1	16	yes	complemented
24	C₂²×D₄×C₆₇	—	2	6	no	—
25	C₂²×D₄×C₆₇	—	2	6	no	—
26	C₂²×D₄×C₆₇	Q₈	1	4	yes	complemented
27	C₂³×C₄×C₆₇	—	2	12	no	—
28	C₂³×C₄×C₆₇	—	2	24	no	—
29	C₂³×C₄×C₆₇	D₄	1	12	yes	—
30	C₂³×C₄×C₆₇	C₂³	1	6	yes	—
31	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
32	C₂³×C₄×C₆₇	C₂³	1	6	yes	—
33	C₂×C₂²⋊C₄×C₆₇	—	2	6	no	—
34	C₂×C₂²⋊C₄×C₆₇	—	2	12	no	—
35	C₂×C₂²⋊C₄×C₆₇	—	2	12	no	—
36	C₂×C₂²⋊C₄×C₆₇	—	2	24	no	—
37	#95×C₆₇	—	2	12	no	—
38	#95×C₆₇	—	2	24	no	—
39	C₂⁴×C₆₇	—	2	4	no	—
40	C₂⁴×C₆₇	—	2	4	no	—
41	C₂⁴×C₆₇	—	2	4	no	—
42	C₂⁴×C₆₇	—	2	3	no	—
43	C₂⁴×C₆₇	—	2	3	no	—
44	C₂⁴×C₆₇	—	2	4	no	—
45	C₂⁴×C₆₇	C₂×D₄	1	6	yes	—
46	C₂⁴×C₆₇	C₂×Q₈	1	2	yes	—
47	C₂⁴×C₆₇	C₂×Q₈	1	2	yes	complemented
48	C₂⁴×C₆₇	#41	1	6	yes	—
49	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
50	C₂×Q₈×C₆₇	—	2	256	no	—
51	C₂×Q₈×C₆₇	—	2	32	no	—
52	C₂×Q₈×C₆₇	—	2	96	no	—
53	C₂×Q₈×C₆₇	—	2	96	no	—
54	C₂×D₄×C₆₇	—	2	24	no	—
55	C₂×D₄×C₆₇	—	4	4	no	—
56	C₂×D₄×C₆₇	—	4	12	no	—
57	C₂²×C₄×C₆₇	—	2	96	no	—
58	C₂²×C₄×C₆₇	—	2	24	no	—
59	C₂²×C₄×C₆₇	—	2	12	no	—
60	C₂²×C₄×C₆₇	—	2	12	no	—
61	C₂²×C₄×C₆₇	—	2	48	no	—
62	C₂²×C₄×C₆₇	—	2	3	no	—
63	C₂²×C₄×C₆₇	—	2	3	no	—
64	C₂²×C₄×C₆₇	—	2	6	no	—
65	C₂²×C₄×C₆₇	—	2	12	no	—
66	C₂²×C₄×C₆₇	—	2	12	no	—
67	C₂²×C₄×C₆₇	—	2	12	no	—
68	C₂²×C₄×C₆₇	C₂×D₄	1	24	yes	—
69	C₂²×C₄×C₆₇	C₂×D₄	1	12	yes	—
70	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	complemented
71	C₂²⋊C₄×C₆₇	—	4	12	no	—
72	C₂²⋊C₄×C₆₇	—	4	24	no	—
73	C₂³×C₆₇	—	2	12	no	—
74	C₂³×C₆₇	—	2	12	no	—
75	C₂³×C₆₇	—	2	24	no	—
76	C₂³×C₆₇	—	2	8	no	—
77	C₂³×C₆₇	—	2	6	no	—
78	C₂³×C₆₇	—	2	8	no	—
79	C₂³×C₆₇	—	4	6	no	—
80	C₂³×C₆₇	—	4	1	no	—
81	C₂³×C₆₇	—	4	3	no	—
82	C₂³×C₆₇	—	4	6	no	—
83	C₂³×C₆₇	—	4	12	no	—
84	C₂³×C₆₇	#120	1	4	yes	—
85	C₂³×C₆₇	C₂²⋊Q₈	1	12	yes	—
86	C₂³×C₆₇	#124	1	12	yes	—
87	C₂³×C₆₇	C₂²×D₄	1	3	yes	—
88	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
89	C₂³×C₆₇	C₂²×Q₈	1	1	yes	characteristic
90	C₂³×C₆₇	C₂×#41	1	3	yes	—
91	C₂³×C₆₇	2₋¹⁺⁴	1	4	yes	—
92	Q₈×C₆₇	—	2	128	no	—
93	D₄×C₆₇	—	4	16	no	—
94	C₂×C₄×C₆₇	—	2	96	no	—
95	C₂×C₄×C₆₇	—	2	12	no	—
96	C₂×C₄×C₆₇	—	2	48	no	—
97	C₂×C₄×C₆₇	—	2	12	no	—
98	C₂×C₄×C₆₇	—	2	6	no	—
99	C₂×C₄×C₆₇	—	4	1	no	—
100	C₂×C₄×C₆₇	—	4	3	no	—
101	C₂×C₄×C₆₇	—	4	6	no	—
102	C₂×C₄×C₆₇	—	4	6	no	—
103	C₂×C₄×C₆₇	—	4	6	no	—
104	C₂×C₄×C₆₇	—	4	24	no	—
105	C₂×C₄×C₆₇	#120	1	24	yes	—
106	C₂²×C₆₇	—	2	4	no	—
107	C₂²×C₆₇	—	2	4	no	—
108	C₂²×C₆₇	—	2	24	no	—
109	C₂²×C₆₇	—	2	24	no	—
110	C₂²×C₆₇	—	4	2	no	—
111	C₂²×C₆₇	—	4	12	no	—
112	C₂²×C₆₇	—	8	1	no	—
113	C₂²×C₆₇	—	8	1	no	—
114	C₂²×C₆₇	—	8	6	no	—
115	C₂²×C₆₇	#393	1	8	yes	complemented
116	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
117	C₂²×C₆₇	C₂×#120	1	6	yes	—
118	C₂²×C₆₇	C₂×C₂²⋊Q₈	1	3	yes	—
119	C₂²×C₆₇	C₂×#124	1	3	yes	—
120	C₂²×C₆₇	#536	1	6	yes	—
121	C₂²×C₆₇	#548	1	12	yes	—
122	C₂²×C₆₇	D₄×Q₈	1	4	yes	—
123	C₄×C₆₇	—	2	24	no	—
124	C₄×C₆₇	—	4	2	no	—
125	C₄×C₆₇	—	8	6	no	—
126	#14873	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
127	C₂×C₆₇	—	2	8	no	—
128	C₂×C₆₇	—	8	2	no	—
129	C₂×C₆₇	C₂×#393	1	1	yes	characteristic
130	C₂×C₆₇	#2409	1	6	yes	—
131	C₂×C₆₇	#2436	1	6	yes	—
132	C₂×C₆₇	#2614	1	1	yes	characteristic
133	C₂×C₆₇	#2846	1	1	yes	characteristic
134	C₂⁴×Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
135	C₂×#393	C₂×C₆₇	1	4	yes	complemented
136	C₂×#386	C₂×C₆₇	1	3	yes	—
137	#1446	C₂×C₆₇	1	4	yes	complemented
138	C₂²⋊#95	C₂×C₆₇	1	3	yes	complemented
139	C₆₇	#14873	1	1	yes	characteristic, direct product factor, Sylow
140	C₂³×Q₈	—	2	16	no	—
141	C₂³×Q₈	—	2	12	no	—
142	C₂³×Q₈	C₂²×C₆₇	1	4	yes	—
143	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
144	C₂⁴×C₄	C₂²×C₆₇	1	3	yes	—
145	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	6	yes	—
146	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	3	yes	—
147	#393	—	2	16	no	—
148	#386	—	2	12	no	—
149	C₂×#95	C₂²×C₆₇	1	12	yes	—
150	C₂×#95	C₂²×C₆₇	1	6	yes	—
151	C₂⁵	C₂³×C₆₇	1	1	yes	characteristic
152	C₂⁵	C₂³×C₆₇	1	1	yes	characteristic
153	C₂²×Q₈	—	2	192	no	—
154	C₂²×Q₈	—	2	32	no	—
155	C₂²×Q₈	—	2	24	no	—
156	C₂²×Q₈	—	2	24	no	—
157	C₂²×Q₈	D₄×C₆₇	1	16	yes	complemented
158	C₂²×D₄	—	2	6	no	—
159	C₂²×D₄	—	2	6	no	—
160	C₂²×D₄	Q₈×C₆₇	1	4	yes	complemented
161	C₂³×C₄	—	2	12	no	—
162	C₂³×C₄	—	2	24	no	—
163	C₂³×C₄	D₄×C₆₇	1	12	yes	—
164	C₂³×C₄	C₂³×C₆₇	1	6	yes	—
165	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
166	C₂³×C₄	C₂³×C₆₇	1	6	yes	—
167	C₂×C₂²⋊C₄	—	2	6	no	—
168	C₂×C₂²⋊C₄	—	2	12	no	—
169	C₂×C₂²⋊C₄	—	2	12	no	—
170	C₂×C₂²⋊C₄	—	2	24	no	—
171	#95	—	2	12	no	—
172	#95	—	2	24	no	—
173	C₂⁴	—	2	4	no	—
174	C₂⁴	—	2	4	no	—
175	C₂⁴	—	2	4	no	—
176	C₂⁴	—	2	3	no	—
177	C₂⁴	—	2	3	no	—
178	C₂⁴	—	2	4	no	—
179	C₂⁴	C₂×D₄×C₆₇	1	6	yes	—
180	C₂⁴	C₂×Q₈×C₆₇	1	2	yes	—
181	C₂⁴	C₂×Q₈×C₆₇	1	2	yes	complemented
182	C₂⁴	#41×C₆₇	1	6	yes	—
183	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
184	C₂×Q₈	—	2	256	no	—
185	C₂×Q₈	—	2	32	no	—
186	C₂×Q₈	—	2	96	no	—
187	C₂×Q₈	—	2	96	no	—
188	C₂×D₄	—	2	24	no	—
189	C₂×D₄	—	4	4	no	—
190	C₂×D₄	—	4	12	no	—
191	C₂²×C₄	—	2	96	no	—
192	C₂²×C₄	—	2	24	no	—
193	C₂²×C₄	—	2	12	no	—
194	C₂²×C₄	—	2	12	no	—
195	C₂²×C₄	—	2	48	no	—
196	C₂²×C₄	—	2	3	no	—
197	C₂²×C₄	—	2	3	no	—
198	C₂²×C₄	—	2	12	no	—
199	C₂²×C₄	—	2	6	no	—
200	C₂²×C₄	—	2	12	no	—
201	C₂²×C₄	—	2	12	no	—
202	C₂²×C₄	C₂×D₄×C₆₇	1	24	yes	—
203	C₂²×C₄	C₂×D₄×C₆₇	1	12	yes	—
204	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	complemented
205	C₂²⋊C₄	—	4	12	no	—
206	C₂²⋊C₄	—	4	24	no	—
207	C₂³	—	2	12	no	—
208	C₂³	—	2	12	no	—
209	C₂³	—	2	24	no	—
210	C₂³	—	2	8	no	—
211	C₂³	—	2	6	no	—
212	C₂³	—	2	8	no	—
213	C₂³	—	4	6	no	—
214	C₂³	—	4	1	no	—
215	C₂³	—	4	3	no	—
216	C₂³	—	4	6	no	—
217	C₂³	—	4	12	no	—
218	C₂³	#120×C₆₇	1	4	yes	—
219	C₂³	C₂²⋊Q₈×C₆₇	1	12	yes	—
220	C₂³	#124×C₆₇	1	12	yes	—
221	C₂³	C₂²×D₄×C₆₇	1	3	yes	—
222	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
223	C₂³	C₂²×Q₈×C₆₇	1	1	yes	characteristic
224	C₂³	C₂×#41×C₆₇	1	3	yes	—
225	C₂³	2₋¹⁺⁴×C₆₇	1	4	yes	—
226	Q₈	—	2	128	no	—
227	D₄	—	4	16	no	—
228	C₂×C₄	—	2	96	no	—
229	C₂×C₄	—	2	12	no	—
230	C₂×C₄	—	2	48	no	—
231	C₂×C₄	—	2	12	no	—
232	C₂×C₄	—	2	6	no	—
233	C₂×C₄	—	4	1	no	—
234	C₂×C₄	—	4	3	no	—
235	C₂×C₄	—	4	6	no	—
236	C₂×C₄	—	4	6	no	—
237	C₂×C₄	—	4	6	no	—
238	C₂×C₄	—	4	24	no	—
239	C₂×C₄	#120×C₆₇	1	24	yes	—
240	C₂²	—	2	4	no	—
241	C₂²	—	2	4	no	—
242	C₂²	—	2	24	no	—
243	C₂²	—	2	24	no	—
244	C₂²	—	4	2	no	—
245	C₂²	—	4	12	no	—
246	C₂²	—	8	1	no	—
247	C₂²	—	8	1	no	—
248	C₂²	—	8	6	no	—
249	C₂²	#393×C₆₇	1	8	yes	complemented
250	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
251	C₂²	C₂×#120×C₆₇	1	6	yes	—
252	C₂²	C₂×C₂²⋊Q₈×C₆₇	1	3	yes	—
253	C₂²	C₂×#124×C₆₇	1	3	yes	—
254	C₂²	#536×C₆₇	1	6	yes	—
255	C₂²	#548×C₆₇	1	12	yes	—
256	C₂²	D₄×Q₈×C₆₇	1	4	yes	—
257	C₄	—	2	24	no	—
258	C₄	—	4	2	no	—
259	C₄	—	8	6	no	—
260	C₂	—	2	8	no	—
261	C₂	—	8	2	no	—
262	C₂	C₂×#393×C₆₇	1	1	yes	characteristic
263	C₂	C₆₇×#2409	1	6	yes	—
264	C₂	C₆₇×#2436	1	6	yes	—
265	C₂	C₆₇×#2614	1	1	yes	characteristic
266	C₂	C₆₇×#2846	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₆₇	C₂⁴
commutator, Frattini, cosocle	C₂⁴	C₂⁴×C₆₇
prime radical	C₂³×D₄×C₆₇	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂⁴×Q₈×C₆₇	C₂	semidirect
2. C₂×#393×C₆₇	C₂	semidirect
4. C₆₇×#1446	C₂	semidirect
5. C₆₇×C₂²⋊#95	C₂	semidirect
23. C₂²×Q₈×C₆₇	D₄	semidirect
26. C₂²×D₄×C₆₇	Q₈	semidirect
47. C₂⁴×C₆₇	C₂×Q₈	semidirect
70. C₂²×C₄×C₆₇	C₂×Q₈	semidirect
115. C₂²×C₆₇	#393	semidirect
126. #14873	C₆₇	direct (N × H)
134. C₂⁴×Q₈	C₂×C₆₇	semidirect
135. C₂×#393	C₂×C₆₇	semidirect
137. #1446	C₂×C₆₇	semidirect
138. C₂²⋊#95	C₂×C₆₇	semidirect
139. C₆₇	#14873	direct (N × H)
157. C₂²×Q₈	D₄×C₆₇	semidirect
160. C₂²×D₄	Q₈×C₆₇	semidirect
181. C₂⁴	C₂×Q₈×C₆₇	semidirect
204. C₂²×C₄	C₂×Q₈×C₆₇	semidirect
249. C₂²	#393×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂⁴×Q₈×C₆₇	2
C₂×#393×C₆₇	2
C₂×#386×C₆₇	2
C₆₇×#1446	2
C₆₇×C₂²⋊#95	2
#14873	67

Subgroups — C67×#14873 #30694124

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#14873 #30694124