Subgroups — #27158490

Order 12416.

492 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41×#87604	C₂	1	2	yes	maximal, complemented
2	#20261776	C₂	1	2	yes	maximal, complemented
3	#20261775	C₂	1	2	yes	maximal, complemented
4	#20261774	C₂	1	4	yes	maximal, complemented
5	#20261745	C₂	1	1	yes	maximal, characteristic, complemented
6	#20261725	C₂	1	2	yes	maximal, complemented
7	C₂×#15690864	C₂	1	1	yes	maximal, characteristic, complemented
8	#135×D₉₇	C₂	1	1	yes	maximal, characteristic
9	C₂×#13095519	C₂²	1	1	yes	characteristic
10	Q₈×#87604	C₂²	1	4	yes	—
11	D₄×#87604	—	2	4	no	—
12	D₄×#87604	C₂²	1	4	yes	—
13	#15690889	—	2	2	no	—
14	#15690889	C₂²	1	2	yes	—
15	C₂×C₄×#87604	—	2	2	no	—
16	C₂×C₄×#87604	C₂²	1	1	yes	characteristic
17	#15690879	C₂²	1	2	yes	—
18	#15690878	C₂²	1	2	yes	—
19	#15690873	C₂²	1	4	yes	—
20	#15690872	C₂²	1	4	yes	—
21	#15690864	C₂²	1	2	yes	—
22	#15690864	C₂²	1	2	yes	—
23	#41×D₉₇	C₂²	1	2	yes	—
24	#15690833	C₄	1	1	yes	characteristic, complemented
25	#15690832	C₄	1	1	yes	characteristic, complemented
26	#15690831	C₄	1	2	yes	complemented
27	#15690830	C₄	1	2	yes	complemented
28	#15690829	C₄	1	1	yes	characteristic, complemented
29	Q₁₆×D₉₇	C₂²	1	1	yes	characteristic
30	SD₁₆×D₉₇	C₂²	1	2	yes	—
31	D₈×D₉₇	C₂²	1	1	yes	characteristic
32	C₂×C₈×D₉₇	C₂²	1	1	yes	characteristic
33	#135×C₉₇	C₄	1	1	yes	characteristic, complemented
34	C₂²×#87604	—	2	4	no	—
35	C₂²×#87604	D₄	1	2	yes	—
36	#13095519	—	2	1	no	—
37	#13095519	C₂³	1	2	yes	—
38	C₂×#11715897	—	2	1	no	—
39	C₄⋊#87604	—	2	4	no	—
40	C₄⋊#87604	C₂³	1	4	yes	—
41	C₂²⋊#87604	—	2	2	no	—
42	C₂²⋊#87604	—	2	4	no	—
43	C₄×#87604	—	2	4	no	—
44	C₄×#87604	—	2	2	no	—
45	C₄×#87604	D₄	1	2	yes	—
46	C₄×#87604	C₂³	1	2	yes	—
47	#13095512	C₂×C₄	1	2	yes	complemented
48	#13095511	—	2	2	no	—
49	#13095511	C₂×C₄	1	2	yes	complemented
50	#13095510	—	2	2	no	—
51	#13095510	C₂×C₄	1	2	yes	complemented
52	Q₈×D₉₇	C₂³	1	2	yes	—
53	D₄×D₉₇	—	2	2	no	—
54	D₄×D₉₇	C₂³	1	2	yes	—
55	C₂×C₄×D₉₇	—	2	2	no	—
56	C₂×C₄×D₉₇	C₂³	1	1	yes	characteristic
57	Dic₃₈₈	C₂×C₄	1	1	yes	characteristic, complemented
58	#13095501	C₂×C₄	1	2	yes	complemented
59	#13095500	C₂×C₄	1	2	yes	complemented
60	#13095499	C₂×C₄	1	2	yes	complemented
61	#13095498	C₂×C₄	1	2	yes	complemented
62	D₇₇₆	C₂×C₄	1	1	yes	characteristic, complemented
63	#13095496	C₂×C₄	1	2	yes	complemented
64	C₂×#11715895	C₂×C₄	1	1	yes	characteristic, complemented
65	C₈×D₉₇	C₂×C₄	1	1	yes	characteristic, complemented
66	C₈×D₉₇	C₂×C₄	1	1	yes	characteristic, complemented
67	C₈×D₉₇	C₂³	1	1	yes	characteristic
68	C₈×D₉₇	C₂³	1	1	yes	characteristic
69	#41×C₉₇	C₂×C₄	1	2	yes	complemented
70	Q₁₆×C₉₇	C₂×C₄	1	1	yes	characteristic, complemented
71	SD₁₆×C₉₇	C₂×C₄	1	2	yes	complemented
72	D₈×C₉₇	C₂×C₄	1	1	yes	characteristic, complemented
73	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic, complemented
74	C₂²×D₉₇	—	2	2	no	—
75	C₂²×D₉₇	C₂×D₄	1	1	yes	characteristic
76	C₂×#87604	—	2	4	no	—
77	C₂×#87604	—	4	2	no	—
78	C₂×#87604	—	4	8	no	—
79	C₂×#87604	C₂×D₄	1	4	yes	—
80	Q₈×C₉₇	C₂²×C₄	1	2	yes	—
81	D₄×C₉₇	—	2	2	no	—
82	D₄×C₉₇	C₂²×C₄	1	2	yes	—
83	C₂×C₄×C₉₇	—	2	2	no	—
84	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
85	C₂²⋊D₉₇	—	2	2	no	—
86	C₂²⋊D₉₇	—	2	2	no	—
87	C₂²⋊D₉₇	—	2	2	no	—
88	C₂×Dic₉₇	—	2	2	no	—
89	C₂×Dic₉₇	#41	1	1	yes	characteristic
90	D₃₈₈	—	2	2	no	—
91	D₃₈₈	C₂²×C₄	1	2	yes	—
92	C₄×D₉₇	—	2	2	no	—
93	C₄×D₉₇	—	2	2	no	—
94	C₄×D₉₇	—	2	2	no	—
95	C₄×D₉₇	C₂²×C₄	1	1	yes	characteristic
96	C₄×D₉₇	C₂×D₄	1	1	yes	characteristic
97	C₄×D₉₇	#41	1	1	yes	characteristic
98	C₄×D₉₇	C₂⁴	1	1	yes	characteristic
99	Dic₁₉₄	—	2	2	no	—
100	Dic₁₉₄	C₂²×C₄	1	2	yes	—
101	#11715897	—	2	2	no	—
102	C₈×C₉₇	C₂²×C₄	1	1	yes	characteristic
103	C₈×C₉₇	C₂²×C₄	1	1	yes	characteristic
104	#11715895	C₂²×C₄	1	1	yes	characteristic
105	#11715895	C₂²×C₄	1	1	yes	characteristic
106	C₂²×C₉₇	—	2	2	no	—
107	C₂²×C₉₇	C₄×D₄	1	1	yes	characteristic
108	C₂×D₉₇	—	2	2	no	—
109	C₂×D₉₇	—	2	2	no	—
110	C₂×D₉₇	—	4	4	no	—
111	C₂×D₉₇	C₄×D₄	1	1	yes	characteristic
112	C₂×D₉₇	C₂²×D₄	1	1	yes	characteristic
113	#87604	—	2	4	no	—
114	#87604	—	4	4	no	—
115	C₄×C₉₇	—	2	2	no	—
116	C₄×C₉₇	C₄×D₄	1	1	yes	characteristic
117	C₄×C₉₇	C₂³×C₄	1	1	yes	characteristic
118	Dic₉₇	—	2	2	no	—
119	Dic₉₇	C₄×D₄	1	1	yes	characteristic
120	Dic₉₇	C₂×#41	1	1	yes	characteristic
121	C₂×C₉₇	—	2	1	no	—
122	C₂×C₉₇	—	4	2	no	—
123	C₂×C₉₇	C₂×C₄×D₄	1	1	yes	characteristic
124	D₉₇	—	2	1	no	—
125	D₉₇	—	4	2	no	—
126	D₉₇	#575	1	2	yes	—
127	#2943	—	97	1	no	maximal, Sylow
128	C₉₇	#2943	1	1	yes	characteristic, complemented, Sylow
129	C₂×#135	—	97	1	no	—
130	C₄×#41	—	97	2	no	—
131	#442	—	97	2	no	—
132	#441	—	97	2	no	—
133	#440	—	97	4	no	—
134	#427	—	97	1	no	—
135	#416	—	97	2	no	—
136	C₂×#97	—	97	1	no	—
137	C₂×#41	—	97	2	no	—
138	#135	—	97	2	no	—
139	#135	—	97	1	no	—
140	#135	—	97	2	no	—
141	#135	—	97	1	no	—
142	#135	—	97	1	no	—
143	#135	#87604	1	1	yes	characteristic, complemented
144	C₂×Q₁₆	—	97	1	no	—
145	C₂×SD₁₆	—	97	2	no	—
146	C₂×D₈	—	97	1	no	—
147	C₂²×C₈	—	97	1	no	—
148	C₂²×C₈	—	97	1	no	—
149	C₄×Q₈	—	97	4	no	—
150	C₄×D₄	—	97	4	no	—
151	C₄×D₄	—	194	4	no	—
152	#117	—	97	2	no	—
153	#117	—	194	2	no	—
154	C₂×C₄²	—	97	1	no	—
155	C₂×C₄²	—	194	2	no	—
156	#107	—	97	2	no	—
157	#106	—	97	2	no	—
158	#103	—	97	4	no	—
159	#102	—	97	4	no	—
160	#97	—	97	2	no	—
161	#97	—	97	2	no	—
162	#41	—	97	2	no	—
163	#41	—	97	2	no	—
164	#41	—	97	2	no	—
165	#41	—	194	2	no	—
166	#41	—	194	2	no	—
167	#41	C₂×#87604	1	2	yes	complemented
168	C₂×Q₈	—	97	2	no	—
169	C₂×D₄	—	97	2	no	—
170	C₂×D₄	—	194	2	no	—
171	C₂²×C₄	—	97	1	no	—
172	C₂²×C₄	—	97	2	no	—
173	C₂²×C₄	—	194	2	no	—
174	C₂²×C₄	—	194	4	no	—
175	Q₁₆	—	97	2	no	—
176	Q₁₆	—	97	1	no	—
177	Q₁₆	C₂×#87604	1	1	yes	characteristic, complemented
178	SD₁₆	—	97	2	no	—
179	SD₁₆	—	97	2	no	—
180	SD₁₆	—	97	2	no	—
181	SD₁₆	C₂×#87604	1	2	yes	complemented
182	D₈	—	97	2	no	—
183	D₈	—	97	1	no	—
184	D₈	C₂×#87604	1	1	yes	characteristic, complemented
185	C₂×C₈	—	97	1	no	—
186	C₂×C₈	—	97	1	no	—
187	C₂×C₈	—	97	1	no	—
188	C₂×C₈	—	97	1	no	—
189	C₂×C₈	—	97	1	no	—
190	C₂×C₈	—	97	2	no	—
191	C₂×C₈	—	194	1	no	—
192	C₂×C₈	—	194	1	no	—
193	C₂×C₈	C₂×#87604	1	1	yes	characteristic, complemented
194	C₄⋊C₄	—	97	4	no	—
195	C₄⋊C₄	—	194	4	no	—
196	C₂²⋊C₄	—	194	4	no	—
197	C₂²⋊C₄	—	194	2	no	—
198	C₄²	—	97	2	no	—
199	C₄²	—	97	2	no	—
200	C₄²	—	194	2	no	—
201	C₄²	—	194	4	no	—
202	C₂³	—	97	1	no	—
203	C₂³	—	194	2	no	—
204	Q₈	—	97	2	no	—
205	Q₈	—	194	2	no	—
206	Q₈	C₂²×#87604	1	2	yes	—
207	D₄	—	2	2	no	—
208	D₄	—	97	2	no	—
209	D₄	—	194	2	no	—
210	D₄	—	194	2	no	—
211	D₄	—	194	2	no	—
212	D₄	—	194	2	no	—
213	D₄	C₂²×#87604	1	2	yes	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	97	1	no	—
216	C₂×C₄	—	97	1	no	—
217	C₂×C₄	—	97	1	no	—
218	C₂×C₄	—	97	1	no	—
219	C₂×C₄	—	97	1	no	—
220	C₂×C₄	—	97	4	no	—
221	C₂×C₄	—	388	8	no	—
222	C₂×C₄	—	388	2	no	—
223	C₂×C₄	—	194	2	no	—
224	C₂×C₄	—	194	2	no	—
225	C₂×C₄	—	194	2	no	—
226	C₂×C₄	—	194	4	no	—
227	C₂×C₄	—	194	2	no	—
228	C₂×C₄	C₂²×#87604	1	1	yes	characteristic
229	C₈	—	97	1	no	—
230	C₈	—	97	1	no	—
231	C₈	—	194	2	no	—
232	C₈	C₂²×#87604	1	1	yes	characteristic
233	C₈	C₂²×#87604	1	1	yes	characteristic
234	C₂²	—	2	2	no	—
235	C₂²	—	97	1	no	—
236	C₂²	—	97	1	no	—
237	C₂²	—	388	4	no	—
238	C₂²	—	194	2	no	—
239	C₂²	—	194	2	no	—
240	C₂²	D₄×#87604	1	1	yes	characteristic
241	C₄	—	2	2	no	—
242	C₄	—	97	1	no	—
243	C₄	—	97	1	no	—
244	C₄	—	388	4	no	—
245	C₄	—	194	4	no	—
246	C₄	—	194	2	no	—
247	C₄	D₄×#87604	1	1	yes	characteristic
248	C₄	C₂³×#87604	1	1	yes	characteristic
249	C₂	—	2	1	no	—
250	C₂	—	4	2	no	—
251	C₂	—	97	2	no	—
252	C₂	—	388	2	no	—
253	C₂	—	194	1	no	—
254	C₂	C₂×D₄×#87604	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×#87604
commutator	C₄×C₉₇	C₂³×C₄
Frattini	C₄	C₂³×#87604
Fitting	#135×C₉₇	C₄
socle	C₂×C₉₇	C₂×C₄×D₄
prime radical	#135×D₉₇	C₂
cosocle	C₄×D₉₇	C₂⁴
hypercenter	#135	#87604
hypocenter	C₉₇	#2943
LCS	G ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #135

Complements

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

N	H ≅ G/N	type
1. #41×#87604	C₂	semidirect
2. #20261776	C₂	semidirect
3. #20261775	C₂	semidirect
4. #20261774	C₂	semidirect
5. #20261745	C₂	semidirect
6. #20261725	C₂	semidirect
7. C₂×#15690864	C₂	semidirect
24. #15690833	C₄	semidirect
25. #15690832	C₄	semidirect
26. #15690831	C₄	semidirect
27. #15690830	C₄	semidirect
28. #15690829	C₄	semidirect
33. #135×C₉₇	C₄	semidirect
47. #13095512	C₂×C₄	semidirect
49. #13095511	C₂×C₄	semidirect
51. #13095510	C₂×C₄	semidirect
57. Dic₃₈₈	C₂×C₄	semidirect
58. #13095501	C₂×C₄	semidirect
59. #13095500	C₂×C₄	semidirect
60. #13095499	C₂×C₄	semidirect
61. #13095498	C₂×C₄	semidirect
62. D₇₇₆	C₂×C₄	semidirect
63. #13095496	C₂×C₄	semidirect
64. C₂×#11715895	C₂×C₄	semidirect
65. C₈×D₉₇	C₂×C₄	semidirect
66. C₈×D₉₇	C₂×C₄	semidirect
69. #41×C₉₇	C₂×C₄	semidirect
70. Q₁₆×C₉₇	C₂×C₄	semidirect
71. SD₁₆×C₉₇	C₂×C₄	semidirect
72. D₈×C₉₇	C₂×C₄	semidirect
73. C₂×C₈×C₉₇	C₂×C₄	semidirect
128. C₉₇	#2943	semidirect
143. #135	#87604	semidirect
167. #41	C₂×#87604	semidirect
177. Q₁₆	C₂×#87604	semidirect
181. SD₁₆	C₂×#87604	semidirect
184. D₈	C₂×#87604	semidirect
193. C₂×C₈	C₂×#87604	semidirect

Maximal subgroups

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

subgroup	index
#41×#87604	2
#20261776	2
#20261775	2
#20261774	2
#20261745	2
#20261725	2
C₂×#15690864	2
#135×D₉₇	2
#2943	97