Subgroups — #27158513

Order 12416.

452 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261814	C₂	1	2	yes	maximal, complemented
2	#20261780	C₂	1	4	yes	maximal, complemented
3	#20261779	C₂	1	2	yes	maximal, complemented
4	#20261777	C₂	1	2	yes	maximal, complemented
5	C₂×#15690881	C₂	1	1	yes	maximal, characteristic, complemented
6	#20261732	C₂	1	2	yes	maximal, complemented
7	#20261698	C₂	1	1	yes	maximal, characteristic, complemented
8	#135×D₉₇	C₂	1	1	yes	maximal, characteristic
9	#15690898	C₂²	1	4	yes	complemented
10	#15690897	—	2	4	no	—
11	#15690897	C₂²	1	4	yes	complemented
12	C₂×#13095521	—	2	2	no	—
13	C₂×#13095521	C₂²	1	2	yes	—
14	D₉₇⋊M₁₆	—	2	2	no	—
15	D₉₇⋊M₁₆	C₂²	1	1	yes	characteristic
16	#15690889	C₂²	1	1	yes	characteristic
17	#15690881	C₂²	1	2	yes	complemented
18	#15690881	C₂²	1	2	yes	complemented
19	#15690875	C₂²	1	4	yes	complemented
20	#15690874	C₂²	1	4	yes	complemented
21	#15690864	C₂²	1	2	yes	complemented
22	C₈×#87604	C₂²	1	2	yes	complemented
23	#41×D₉₇	C₂²	1	2	yes	—
24	#15690833	C₂²	1	1	yes	characteristic
25	#15690832	C₄	1	1	yes	characteristic, complemented
26	#15690831	C₂²	1	2	yes	—
27	#15690830	C₄	1	2	yes	complemented
28	#15690829	C₂²	1	1	yes	characteristic
29	Q₁₆×D₉₇	C₄	1	1	yes	characteristic, complemented
30	SD₁₆×D₉₇	C₄	1	2	yes	complemented
31	D₈×D₉₇	C₄	1	1	yes	characteristic, complemented
32	C₂×C₈×D₉₇	C₂²	1	1	yes	characteristic
33	#135×C₉₇	C₄	1	1	yes	characteristic, complemented
34	#13095521	—	2	2	no	—
35	#13095521	—	2	4	no	—
36	#13095521	D₄	1	1	yes	characteristic
37	#13095521	D₄	1	1	yes	characteristic
38	#13095521	C₂³	1	4	yes	—
39	#13095520	—	2	4	no	—
40	#13095520	D₄	1	2	yes	—
41	#13095519	—	2	4	no	—
42	#13095519	C₂³	1	2	yes	—
43	C₂×#11715897	—	2	2	no	—
44	C₂×#11715897	—	2	4	no	—
45	C₄⋊#87604	—	2	1	no	—
46	C₂²⋊#87604	—	2	1	no	—
47	C₄×#87604	C₂³	1	2	yes	—
48	#13095512	C₂³	1	2	yes	—
49	#13095511	—	2	2	no	—
50	#13095511	C₂×C₄	1	2	yes	—
51	#13095510	—	2	2	no	—
52	#13095510	C₂³	1	2	yes	—
53	Q₈×D₉₇	C₂×C₄	1	2	yes	—
54	D₄×D₉₇	—	2	2	no	—
55	D₄×D₉₇	C₂×C₄	1	2	yes	—
56	C₂×C₄×D₉₇	—	2	2	no	—
57	C₂×C₄×D₉₇	C₂³	1	1	yes	characteristic
58	Dic₃₈₈	C₂×C₄	1	1	yes	characteristic
59	#13095501	C₂×C₄	1	2	yes	—
60	#13095500	C₂×C₄	1	2	yes	—
61	#13095499	C₂×C₄	1	2	yes	—
62	#13095498	C₂×C₄	1	2	yes	—
63	D₇₇₆	C₂×C₄	1	1	yes	characteristic
64	#13095496	C₂×C₄	1	2	yes	—
65	C₂×#11715895	C₂×C₄	1	1	yes	characteristic
66	C₈×D₉₇	C₂×C₄	1	1	yes	characteristic
67	C₈×D₉₇	C₂×C₄	1	1	yes	characteristic
68	C₈×D₉₇	C₂³	1	1	yes	characteristic
69	C₈×D₉₇	C₂³	1	1	yes	characteristic
70	#41×C₉₇	C₂×C₄	1	2	yes	—
71	Q₁₆×C₉₇	C₂×C₄	1	1	yes	characteristic
72	SD₁₆×C₉₇	C₂×C₄	1	2	yes	—
73	D₈×C₉₇	C₂×C₄	1	1	yes	characteristic
74	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
75	C₂²×D₉₇	—	2	2	no	—
76	C₂²×D₉₇	#41	1	1	yes	characteristic
77	C₂×#87604	—	2	2	no	—
78	Q₈×C₉₇	C₂²×C₄	1	2	yes	—
79	D₄×C₉₇	—	2	2	no	—
80	D₄×C₉₇	C₂²×C₄	1	2	yes	—
81	C₂×C₄×C₉₇	—	2	2	no	—
82	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
83	C₂²⋊D₉₇	—	2	2	no	—
84	C₂²⋊D₉₇	—	2	2	no	—
85	C₂²⋊D₉₇	—	2	2	no	—
86	C₂×Dic₉₇	—	2	2	no	—
87	C₂×Dic₉₇	C₂×D₄	1	1	yes	characteristic
88	D₃₈₈	—	2	2	no	—
89	D₃₈₈	C₂²×C₄	1	2	yes	—
90	C₄×D₉₇	—	2	2	no	—
91	C₄×D₉₇	—	2	2	no	—
92	C₄×D₉₇	—	2	2	no	—
93	C₄×D₉₇	C₂²×C₄	1	1	yes	characteristic
94	C₄×D₉₇	C₂×D₄	1	1	yes	characteristic
95	C₄×D₉₇	#41	1	1	yes	characteristic
96	C₄×D₉₇	C₂⁴	1	1	yes	characteristic
97	Dic₁₉₄	—	2	2	no	—
98	Dic₁₉₄	C₂²×C₄	1	2	yes	—
99	#11715897	—	2	4	no	—
100	#11715897	C₂×D₄	1	2	yes	—
101	#11715897	C₂×D₄	1	2	yes	—
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	—	4	1	no	—
110	C₂×D₉₇	—	8	2	no	—
111	C₂×D₉₇	C₄×D₄	1	1	yes	characteristic
112	C₂×D₉₇	C₂×#41	1	1	yes	characteristic
113	#87604	—	4	2	no	—
114	C₄×C₉₇	—	2	2	no	—
115	C₄×C₉₇	C₄×D₄	1	1	yes	characteristic
116	C₄×C₉₇	C₂³×C₄	1	1	yes	characteristic
117	Dic₉₇	—	2	2	no	—
118	Dic₉₇	C₄×D₄	1	1	yes	characteristic
119	Dic₉₇	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₉₇	—	2	1	no	—
121	C₂×C₉₇	—	4	2	no	—
122	C₂×C₉₇	C₂×C₄×D₄	1	1	yes	characteristic
123	D₉₇	—	2	1	no	—
124	D₉₇	—	2	1	no	—
125	D₉₇	—	4	2	no	—
126	#2957	—	97	1	no	maximal, Sylow
127	C₉₇	#2957	1	1	yes	characteristic, complemented, Sylow
128	C₂×#135	—	97	1	no	—
129	#568	—	97	2	no	—
130	#444	—	97	4	no	—
131	#443	—	97	2	no	—
132	#443	—	97	2	no	—
133	C₂×#108	—	97	1	no	—
134	#421	—	97	2	no	—
135	#405	—	97	1	no	—
136	C₂×#41	—	97	2	no	—
137	#135	—	97	2	no	—
138	#135	—	97	1	no	—
139	#135	—	97	2	no	—
140	#135	—	97	1	no	—
141	#135	—	97	1	no	—
142	#135	#87604	1	1	yes	characteristic, complemented
143	C₂×Q₁₆	—	97	1	no	—
144	C₂×SD₁₆	—	97	2	no	—
145	C₂×D₈	—	97	1	no	—
146	#131	—	97	4	no	—
147	#131	—	97	4	no	—
148	#131	—	194	4	no	—
149	C₂×M₁₆	—	97	1	no	—
150	C₂×M₁₆	—	97	2	no	—
151	C₂×M₁₆	—	194	2	no	—
152	C₂×M₁₆	—	194	2	no	—
153	C₂²×C₈	—	97	1	no	—
154	#117	—	97	1	no	—
155	#108	—	97	2	no	—
156	#108	—	97	2	no	—
157	#104	—	97	4	no	—
158	#104	—	97	4	no	—
159	#97	—	97	2	no	—
160	C₄×C₈	—	97	2	no	—
161	#41	—	97	2	no	—
162	#41	—	97	2	no	—
163	#41	—	97	2	no	—
164	#41	—	194	2	no	—
165	#41	—	194	2	no	—
166	#41	C₂×#87604	1	2	yes	—
167	C₂×Q₈	—	97	2	no	—
168	C₂×D₄	—	97	2	no	—
169	C₂×D₄	—	194	2	no	—
170	C₂²×C₄	—	97	1	no	—
171	C₂²×C₄	—	194	2	no	—
172	Q₁₆	—	97	2	no	—
173	Q₁₆	—	97	1	no	—
174	Q₁₆	C₂×#87604	1	1	yes	characteristic
175	SD₁₆	—	97	2	no	—
176	SD₁₆	—	97	2	no	—
177	SD₁₆	—	97	2	no	—
178	SD₁₆	C₂×#87604	1	2	yes	—
179	D₈	—	97	2	no	—
180	D₈	—	97	1	no	—
181	D₈	C₂×#87604	1	1	yes	characteristic
182	M₁₆	—	97	2	no	—
183	M₁₆	—	97	1	no	—
184	M₁₆	—	97	1	no	—
185	M₁₆	—	97	4	no	—
186	M₁₆	—	194	4	no	—
187	M₁₆	—	194	4	no	—
188	M₁₆	—	194	2	no	—
189	C₂×C₈	—	97	2	no	—
190	C₂×C₈	—	97	1	no	—
191	C₂×C₈	—	97	1	no	—
192	C₂×C₈	—	97	1	no	—
193	C₂×C₈	—	97	1	no	—
194	C₂×C₈	—	97	1	no	—
195	C₂×C₈	—	194	4	no	—
196	C₂×C₈	—	194	4	no	—
197	C₂×C₈	—	194	2	no	—
198	C₂×C₈	C₂×#87604	1	1	yes	characteristic
199	C₄⋊C₄	—	194	1	no	—
200	C₂²⋊C₄	—	194	1	no	—
201	C₄²	—	97	2	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	—	194	2	no	—
221	C₂×C₄	—	194	2	no	—
222	C₂×C₄	—	194	2	no	—
223	C₂×C₄	—	194	2	no	—
224	C₂×C₄	—	194	2	no	—
225	C₂×C₄	C₂²×#87604	1	1	yes	characteristic
226	C₈	—	97	2	no	—
227	C₈	—	97	2	no	—
228	C₈	—	97	1	no	—
229	C₈	—	97	1	no	—
230	C₈	—	194	4	no	—
231	C₈	C₂²×#87604	1	1	yes	characteristic
232	C₈	C₂²×#87604	1	1	yes	characteristic
233	C₂²	—	2	2	no	—
234	C₂²	—	776	2	no	—
235	C₂²	—	97	1	no	—
236	C₂²	—	97	1	no	—
237	C₂²	—	388	1	no	—
238	C₂²	—	194	2	no	—
239	C₂²	D₄×#87604	1	1	yes	characteristic
240	C₄	—	2	2	no	—
241	C₄	—	97	1	no	—
242	C₄	—	97	1	no	—
243	C₄	—	388	2	no	—
244	C₄	—	194	2	no	—
245	C₄	D₄×#87604	1	1	yes	characteristic
246	C₄	C₂³×#87604	1	1	yes	characteristic
247	C₂	—	2	1	no	—
248	C₂	—	4	2	no	—
249	C₂	—	388	2	no	—
250	C₂	—	194	1	no	—
251	C₂	—	194	1	no	—
252	C₂	C₂×D₄×#87604	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×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	#2957
LCS	G ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #135

Complements

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

N	H ≅ G/N	type
1. #20261814	C₂	semidirect
2. #20261780	C₂	semidirect
3. #20261779	C₂	semidirect
4. #20261777	C₂	semidirect
5. C₂×#15690881	C₂	semidirect
6. #20261732	C₂	semidirect
7. #20261698	C₂	semidirect
9. #15690898	C₂²	semidirect
11. #15690897	C₂²	semidirect
17. #15690881	C₂²	semidirect
18. #15690881	C₂²	semidirect
19. #15690875	C₂²	semidirect
20. #15690874	C₂²	semidirect
21. #15690864	C₂²	semidirect
22. C₈×#87604	C₂²	semidirect
25. #15690832	C₄	semidirect
27. #15690830	C₄	semidirect
29. Q₁₆×D₉₇	C₄	semidirect
30. SD₁₆×D₉₇	C₄	semidirect
31. D₈×D₉₇	C₄	semidirect
33. #135×C₉₇	C₄	semidirect
127. C₉₇	#2957	semidirect
142. #135	#87604	semidirect

Maximal subgroups

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

subgroup	index
#20261814	2
#20261780	2
#20261779	2
#20261777	2
C₂×#15690881	2
#20261732	2
#20261698	2
#135×D₉₇	2
#2957	97