Subgroups — #30786991

Order 17152.

6494 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863256	C₂	1	6	yes	maximal, complemented
2	C₂×#16863252	C₂	1	2	yes	maximal
3	#22558432	C₂	1	6	yes	maximal, complemented
4	C₂⁵×C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊Dic₆₇	C₂²	1	4	yes	complemented
6	C₂²×C₂²⋊Dic₆₇	C₂²	1	12	yes	—
7	#16863256	—	2	24	no	—
8	#16863252	—	2	8	no	—
9	C₂×C₆₇⋊#95	C₂²	1	12	yes	complemented
10	C₂⁶×C₆₇	C₂²	1	1	yes	characteristic
11	C₂⁴×C₄×C₆₇	—	2	24	no	—
12	C₂⁴×C₄×C₆₇	C₄	1	8	yes	complemented
13	C₂⁴×C₄×C₆₇	C₂²	1	3	yes	—
14	C₂⁴×C₄×C₆₇	C₂²	1	3	yes	—
15	C₂³×Dic₆₇	C₂³	1	8	yes	—
16	C₂×C₄⋊Dic₆₇	—	2	24	no	—
17	C₂×C₂²⋊Dic₆₇	—	2	24	no	—
18	C₂×C₂²⋊Dic₆₇	—	2	24	no	—
19	C₂×C₂²⋊Dic₆₇	—	2	24	no	—
20	C₆₇⋊#95	—	2	24	no	—
21	C₂⁵×C₆₇	—	2	12	no	—
22	C₂⁵×C₆₇	—	2	12	no	—
23	C₂⁵×C₆₇	C₂×C₄	1	4	yes	—
24	C₂⁵×C₆₇	D₄	1	4	yes	—
25	C₂⁵×C₆₇	Q₈	1	4	yes	—
26	C₂⁵×C₆₇	C₂³	1	3	yes	—
27	C₂³×C₄×C₆₇	—	2	128	no	—
28	C₂³×C₄×C₆₇	—	2	12	no	—
29	C₂³×C₄×C₆₇	—	2	96	no	—
30	C₂³×C₄×C₆₇	—	2	24	no	—
31	C₂³×C₄×C₆₇	—	2	24	no	—
32	C₂³×C₄×C₆₇	C₂×C₄	1	12	yes	—
33	C₂³×C₄×C₆₇	C₂×C₄	1	12	yes	—
34	C₂³×C₄×C₆₇	D₄	1	24	yes	—
35	C₂³×C₄×C₆₇	C₂³	1	3	yes	—
36	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
37	C₂²×Dic₆₇	—	2	24	no	—
38	C₂²×Dic₆₇	—	2	8	no	—
39	C₂²×Dic₆₇	—	2	24	no	—
40	C₄⋊Dic₆₇	—	4	16	no	—
41	C₂²⋊Dic₆₇	—	4	48	no	—
42	C₂⁴×C₆₇	—	2	32	no	—
43	C₂⁴×C₆₇	—	2	96	no	—
44	C₂⁴×C₆₇	—	2	24	no	—
45	C₂⁴×C₆₇	—	2	24	no	—
46	C₂⁴×C₆₇	—	2	48	no	—
47	C₂⁴×C₆₇	—	2	48	no	—
48	C₂⁴×C₆₇	—	2	12	no	—
49	C₂⁴×C₆₇	—	2	12	no	—
50	C₂⁴×C₆₇	C₄⋊C₄	1	16	yes	complemented
51	C₂⁴×C₆₇	C₂²×C₄	1	6	yes	—
52	C₂⁴×C₆₇	C₂×D₄	1	12	yes	—
53	C₂⁴×C₆₇	C₂×D₄	1	6	yes	—
54	C₂⁴×C₆₇	C₂×Q₈	1	6	yes	—
55	C₂⁴×C₆₇	#41	1	12	yes	—
56	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
57	C₂²×C₄×C₆₇	—	2	96	no	—
58	C₂²×C₄×C₆₇	—	2	256	no	—
59	C₂²×C₄×C₆₇	—	2	24	no	—
60	C₂²×C₄×C₆₇	—	2	12	no	—
61	C₂²×C₄×C₆₇	—	2	48	no	—
62	C₂²×C₄×C₆₇	—	2	48	no	—
63	C₂²×C₄×C₆₇	—	2	96	no	—
64	C₂²×C₄×C₆₇	C₂²⋊C₄	1	48	yes	complemented
65	C₂²×C₄×C₆₇	C₂²×C₄	1	6	yes	—
66	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₆₇	C₂×D₄	1	12	yes	—
68	C₂²×C₄×C₆₇	C₂×D₄	1	12	yes	—
69	C₂³×C₆₇	—	2	32	no	—
70	C₂³×C₆₇	—	2	128	no	—
71	C₂³×C₆₇	—	2	128	no	—
72	C₂³×C₆₇	—	2	96	no	—
73	C₂³×C₆₇	—	2	12	no	—
74	C₂³×C₆₇	—	2	12	no	—
75	C₂³×C₆₇	—	2	24	no	—
76	C₂³×C₆₇	—	2	24	no	—
77	C₂³×C₆₇	—	2	24	no	—
78	C₂³×C₆₇	—	2	24	no	—
79	C₂³×C₆₇	—	2	48	no	—
80	C₂³×C₆₇	—	2	96	no	—
81	C₂³×C₆₇	C₂×C₂²⋊C₄	1	12	yes	—
82	C₂³×C₆₇	C₂×C₄⋊C₄	1	12	yes	—
83	C₂³×C₆₇	#117	1	12	yes	—
84	C₂³×C₆₇	#121	1	24	yes	—
85	C₂³×C₆₇	C₂²⋊Q₈	1	24	yes	—
86	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
87	C₂³×C₆₇	C₂²×D₄	1	6	yes	—
88	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
89	C₂³×C₆₇	C₂²×Q₈	1	1	yes	characteristic
90	C₂³×C₆₇	C₂×#41	1	6	yes	—
91	C₂×C₄×C₆₇	—	2	96	no	—
92	C₂×C₄×C₆₇	—	2	12	no	—
93	C₂×C₄×C₆₇	—	2	24	no	—
94	C₂×C₄×C₆₇	—	2	96	no	—
95	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	12	yes	—
96	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	12	yes	—
97	C₂×C₄×C₆₇	#120	1	16	yes	—
98	C₂×Dic₆₇	—	4	8	no	—
99	C₂×Dic₆₇	—	4	24	no	—
100	C₂×Dic₆₇	—	4	24	no	—
101	C₂²×C₆₇	—	2	128	no	—
102	C₂²×C₆₇	—	2	48	no	—
103	C₂²×C₆₇	—	2	48	no	—
104	C₂²×C₆₇	—	2	12	no	—
105	C₂²×C₆₇	—	2	24	no	—
106	C₂²×C₆₇	—	2	12	no	—
107	C₂²×C₆₇	—	2	24	no	—
108	C₂²×C₆₇	#380	1	24	yes	complemented
109	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	3	yes	—
110	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
111	C₂²×C₆₇	C₂×#117	1	3	yes	—
112	C₂²×C₆₇	C₂×#120	1	4	yes	—
113	C₂²×C₆₇	C₂×#121	1	6	yes	—
114	C₂²×C₆₇	C₂×C₂²⋊Q₈	1	6	yes	—
115	C₂²×C₆₇	#525	1	12	yes	—
116	C₄×C₆₇	—	2	12	no	—
117	C₄×C₆₇	#379	1	8	yes	complemented
118	Dic₆₇	—	8	8	no	—
119	#14680	—	67	1	no	maximal, Sylow
120	C₂×C₆₇	—	2	24	no	—
121	C₂×C₆₇	C₂×#379	1	1	yes	characteristic
122	C₂×C₆₇	C₂×#380	1	3	yes	—
123	C₂×C₆₇	#2280	1	3	yes	—
124	C₂×C₆₇	#2847	1	4	yes	—
125	C₂×C₆₇	#2848	1	4	yes	—
126	C₂⁵×C₄	D₆₇	1	1	yes	characteristic
127	C₂×#380	—	67	6	no	—
128	C₂×#379	—	67	2	no	—
129	C₂²⋊#95	—	67	6	no	—
130	C₆₇	#14680	1	1	yes	characteristic, complemented, Sylow
131	C₂⁶	C₂×D₆₇	1	1	yes	characteristic
132	C₂⁴×C₄	—	2	24	no	—
133	C₂⁴×C₄	Dic₆₇	1	8	yes	complemented
134	C₂⁴×C₄	C₂×D₆₇	1	3	yes	—
135	C₂⁴×C₄	C₂×D₆₇	1	3	yes	—
136	C₂²×C₄⋊C₄	—	67	4	no	—
137	C₂²×C₂²⋊C₄	—	67	12	no	—
138	#380	—	134	24	no	—
139	#379	—	134	8	no	—
140	C₂×#95	—	67	12	no	—
141	C₂⁵	—	2	12	no	—
142	C₂⁵	—	2	12	no	—
143	C₂⁵	Dic₁₃₄	1	4	yes	—
144	C₂⁵	D₂₆₈	1	4	yes	—
145	C₂⁵	C₂×Dic₆₇	1	4	yes	—
146	C₂⁵	C₂²×D₆₇	1	3	yes	—
147	C₂³×C₄	—	2	128	no	—
148	C₂³×C₄	—	2	12	no	—
149	C₂³×C₄	—	2	96	no	—
150	C₂³×C₄	—	2	24	no	—
151	C₂³×C₄	—	2	24	no	—
152	C₂³×C₄	—	67	8	no	—
153	C₂³×C₄	C₂×Dic₆₇	1	12	yes	—
154	C₂³×C₄	C₂×Dic₆₇	1	12	yes	—
155	C₂³×C₄	C₂²⋊D₆₇	1	24	yes	—
156	C₂³×C₄	C₂²×D₆₇	1	3	yes	—
157	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
158	C₂×C₄⋊C₄	—	134	24	no	—
159	C₂×C₂²⋊C₄	—	134	24	no	—
160	C₂×C₂²⋊C₄	—	134	24	no	—
161	C₂×C₂²⋊C₄	—	134	24	no	—
162	#95	—	134	24	no	—
163	C₂⁴	—	2	32	no	—
164	C₂⁴	—	2	96	no	—
165	C₂⁴	—	2	24	no	—
166	C₂⁴	—	2	24	no	—
167	C₂⁴	—	2	48	no	—
168	C₂⁴	—	2	48	no	—
169	C₂⁴	—	2	12	no	—
170	C₂⁴	—	2	12	no	—
171	C₂⁴	C₄⋊Dic₆₇	1	16	yes	complemented
172	C₂⁴	C₂²×Dic₆₇	1	6	yes	—
173	C₂⁴	C₂×C₂²⋊D₆₇	1	12	yes	—
174	C₂⁴	C₂×D₂₆₈	1	6	yes	—
175	C₂⁴	C₂×Dic₁₃₄	1	6	yes	—
176	C₂⁴	#11761628	1	12	yes	—
177	C₂⁴	C₂³×D₆₇	1	1	yes	characteristic
178	C₂²×C₄	—	2	96	no	—
179	C₂²×C₄	—	2	256	no	—
180	C₂²×C₄	—	2	24	no	—
181	C₂²×C₄	—	2	12	no	—
182	C₂²×C₄	—	2	48	no	—
183	C₂²×C₄	—	2	48	no	—
184	C₂²×C₄	—	2	96	no	—
185	C₂²×C₄	—	134	24	no	—
186	C₂²×C₄	—	134	8	no	—
187	C₂²×C₄	—	134	24	no	—
188	C₂²×C₄	C₂²⋊Dic₆₇	1	48	yes	complemented
189	C₂²×C₄	C₂²×Dic₆₇	1	6	yes	—
190	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
191	C₂²×C₄	C₂×C₂²⋊D₆₇	1	12	yes	—
192	C₂²×C₄	C₂×C₂²⋊D₆₇	1	12	yes	—
193	C₄⋊C₄	—	268	16	no	—
194	C₂²⋊C₄	—	268	48	no	—
195	C₂³	—	2	32	no	—
196	C₂³	—	2	96	no	—
197	C₂³	—	2	128	no	—
198	C₂³	—	2	128	no	—
199	C₂³	—	2	96	no	—
200	C₂³	—	2	12	no	—
201	C₂³	—	2	12	no	—
202	C₂³	—	2	24	no	—
203	C₂³	—	2	24	no	—
204	C₂³	—	2	24	no	—
205	C₂³	—	2	24	no	—
206	C₂³	—	2	48	no	—
207	C₂³	C₂×C₂²⋊Dic₆₇	1	12	yes	—
208	C₂³	C₂×C₄⋊Dic₆₇	1	12	yes	—
209	C₂³	#14524002	1	12	yes	—
210	C₂³	#14524017	1	24	yes	—
211	C₂³	#14524022	1	24	yes	—
212	C₂³	C₂³×Dic₆₇	1	1	yes	characteristic
213	C₂³	C₂²×C₂²⋊D₆₇	1	6	yes	—
214	C₂³	C₂²×D₂₆₈	1	1	yes	characteristic
215	C₂³	C₂²×Dic₁₃₄	1	1	yes	characteristic
216	C₂³	C₂×#11761628	1	6	yes	—
217	C₂×C₄	—	2	96	no	—
218	C₂×C₄	—	2	12	no	—
219	C₂×C₄	—	2	24	no	—
220	C₂×C₄	—	2	96	no	—
221	C₂×C₄	—	268	8	no	—
222	C₂×C₄	—	268	24	no	—
223	C₂×C₄	—	268	24	no	—
224	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	12	yes	—
225	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	12	yes	—
226	C₂×C₄	#14524012	1	16	yes	—
227	C₂²	—	2	128	no	—
228	C₂²	—	2	12	no	—
229	C₂²	—	2	24	no	—
230	C₂²	—	2	12	no	—
231	C₂²	—	2	24	no	—
232	C₂²	—	2	48	no	—
233	C₂²	—	2	48	no	—
234	C₂²	#16863256	1	24	yes	complemented
235	C₂²	C₂²×C₂²⋊Dic₆₇	1	3	yes	—
236	C₂²	C₂²×C₄⋊Dic₆₇	1	1	yes	characteristic
237	C₂²	C₂×#14524002	1	3	yes	—
238	C₂²	C₂×#14524012	1	4	yes	—
239	C₂²	C₂×#14524017	1	6	yes	—
240	C₂²	C₂×#14524022	1	6	yes	—
241	C₂²	#16863907	1	12	yes	—
242	C₄	—	2	12	no	—
243	C₄	—	536	8	no	—
244	C₄	#16863252	1	8	yes	complemented
245	C₂	—	2	24	no	—
246	C₂	C₂×#16863252	1	1	yes	characteristic
247	C₂	C₂×#16863256	1	3	yes	—
248	C₂	#22562592	1	3	yes	—
249	C₂	#22568491	1	4	yes	—
250	C₂	#22568496	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂⁴	C₂³×D₆₇
commutator	C₂³×C₆₇	C₂³×C₄
Fitting	C₂⁵×C₄×C₆₇	C₂
socle, cosocle	C₂⁴×C₆₇	C₂⁴
prime radical	C₂⁶×C₆₇	C₂²
hypercenter	C₂⁵×C₄	D₆₇
hypocenter	C₆₇	#14680
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂⁴ ◁ C₂⁵×C₄

Complements

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

N	H ≅ G/N	type
1. C₂×#16863256	C₂	semidirect
3. #22558432	C₂	semidirect
5. C₂²×C₄⋊Dic₆₇	C₂²	semidirect
9. C₂×C₆₇⋊#95	C₂²	semidirect
12. C₂⁴×C₄×C₆₇	C₄	semidirect
50. C₂⁴×C₆₇	C₄⋊C₄	semidirect
64. C₂²×C₄×C₆₇	C₂²⋊C₄	semidirect
108. C₂²×C₆₇	#380	semidirect
117. C₄×C₆₇	#379	semidirect
130. C₆₇	#14680	semidirect
133. C₂⁴×C₄	Dic₆₇	semidirect
171. C₂⁴	C₄⋊Dic₆₇	semidirect
188. C₂²×C₄	C₂²⋊Dic₆₇	semidirect
234. C₂²	#16863256	semidirect
244. C₄	#16863252	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863256	2
C₂×#16863252	2
#22558432	2
C₂⁵×C₄×C₆₇	2
#14680	67