Subgroups — #27158496

Order 12416.

492 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261803	C₂	1	1	yes	maximal, characteristic, complemented
2	#20261802	C₂	1	1	yes	maximal, characteristic
3	#20261774	C₂	1	4	yes	maximal, complemented
4	SD₁₆×#87604	C₂	1	4	yes	maximal, complemented
5	#20261745	C₂	1	1	yes	maximal, characteristic, complemented
6	#20261728	C₂	1	1	yes	maximal, characteristic, complemented
7	#20261727	C₂	1	1	yes	maximal, characteristic
8	#20261698	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×SD₁₆×D₉₇	C₂	1	1	yes	maximal, characteristic
10	D₉₇⋊M₁₆	C₂²	1	1	yes	characteristic
11	Q₈×#87604	—	2	2	no	—
12	Q₈×#87604	C₂²	1	4	yes	complemented
13	D₄×#87604	—	2	2	no	—
14	D₄×#87604	C₂²	1	4	yes	—
15	#15690889	—	2	2	no	—
16	#15690889	C₂²	1	1	yes	characteristic
17	#15690889	C₂²	1	1	yes	characteristic
18	#15690889	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊#87604	—	2	2	no	—
20	#15690879	C₂²	1	2	yes	complemented
21	#15690878	C₂²	1	2	yes	complemented
22	#15690873	C₂²	1	4	yes	complemented
23	#15690872	C₂²	1	4	yes	—
24	#15690864	C₂²	1	2	yes	complemented
25	C₈×#87604	C₂²	1	2	yes	complemented
26	C₂×Q₈×D₉₇	C₂²	1	1	yes	characteristic
27	C₂×D₄×D₉₇	C₂²	1	1	yes	characteristic
28	C₂×#13095500	C₄	1	1	yes	characteristic, complemented
29	C₂×#13095499	C₄	1	1	yes	characteristic, complemented
30	C₂×#13095498	C₄	1	1	yes	characteristic, complemented
31	SD₁₆×D₉₇	C₄	1	2	yes	complemented
32	SD₁₆×D₉₇	C₄	1	2	yes	complemented
33	SD₁₆×D₉₇	C₂²	1	2	yes	—
34	SD₁₆×D₉₇	C₂²	1	2	yes	—
35	C₂×C₈×D₉₇	C₂²	1	1	yes	characteristic
36	C₂×SD₁₆×C₉₇	C₄	1	1	yes	characteristic, complemented
37	C₂²×#87604	—	2	4	no	—
38	#13095521	—	2	1	no	—
39	#13095520	—	2	1	no	—
40	#13095519	C₂³	1	2	yes	—
41	C₄⋊#87604	—	2	4	no	—
42	C₄⋊#87604	D₄	1	2	yes	—
43	C₄⋊#87604	C₂³	1	2	yes	—
44	C₄⋊#87604	C₂³	1	2	yes	—
45	C₂²⋊#87604	—	2	1	no	—
46	C₂²⋊#87604	—	2	1	no	—
47	C₂²⋊#87604	—	2	4	no	—
48	C₂²⋊#87604	D₄	1	2	yes	—
49	C₄×#87604	—	2	4	no	—
50	C₄×#87604	—	2	1	no	—
51	C₄×#87604	—	2	1	no	—
52	C₄×#87604	C₂³	1	2	yes	—
53	C₂³×D₉₇	—	2	1	no	—
54	C₂×Dic₁₉₄	—	2	1	no	—
55	C₂×Dic₁₉₄	C₂×C₄	1	1	yes	characteristic, complemented
56	Q₈×D₉₇	—	2	1	no	—
57	Q₈×D₉₇	—	2	1	no	—
58	Q₈×D₉₇	C₂×C₄	1	2	yes	complemented
59	Q₈×D₉₇	C₂³	1	2	yes	—
60	C₂×D₃₈₈	C₂×C₄	1	1	yes	characteristic
61	C₂×C₂²⋊D₉₇	—	2	1	no	—
62	D₄×D₉₇	—	2	1	no	—
63	D₄×D₉₇	—	2	1	no	—
64	D₄×D₉₇	C₂×C₄	1	2	yes	—
65	D₄×D₉₇	C₂³	1	2	yes	—
66	C₂×C₄×D₉₇	—	2	1	no	—
67	C₂×C₄×D₉₇	C₂³	1	1	yes	characteristic
68	#13095500	C₂×C₄	1	2	yes	complemented
69	#13095500	C₂×C₄	1	2	yes	complemented
70	#13095499	C₂×C₄	1	2	yes	complemented
71	#13095499	C₂×C₄	1	2	yes	complemented
72	#13095498	C₂×C₄	1	2	yes	complemented
73	#13095498	C₂×C₄	1	2	yes	complemented
74	C₂×#11715895	C₂×C₄	1	1	yes	characteristic, complemented
75	C₈×D₉₇	C₂×C₄	1	1	yes	characteristic, complemented
76	C₈×D₉₇	C₂×C₄	1	1	yes	characteristic, complemented
77	C₈×D₉₇	C₂³	1	1	yes	characteristic
78	C₈×D₉₇	C₂³	1	1	yes	characteristic
79	C₂×Q₈×C₉₇	C₂×C₄	1	1	yes	characteristic, complemented
80	C₂×D₄×C₉₇	C₂×C₄	1	1	yes	characteristic
81	SD₁₆×C₉₇	C₂×C₄	1	2	yes	complemented
82	SD₁₆×C₉₇	C₂×C₄	1	2	yes	complemented
83	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic, complemented
84	C₂³×C₉₇	—	2	1	no	—
85	C₂²×D₉₇	—	2	1	no	—
86	C₂²×D₉₇	—	2	2	no	—
87	C₂²×D₉₇	—	2	2	no	—
88	C₂²×D₉₇	—	8	2	no	—
89	C₂²×D₉₇	C₂×D₄	1	1	yes	characteristic
90	C₂×#87604	—	2	2	no	—
91	C₂×#87604	—	2	2	no	—
92	C₂×#87604	—	8	4	no	—
93	C₂×#87604	C₂×D₄	1	4	yes	—
94	Q₈×C₉₇	—	2	1	no	—
95	Q₈×C₉₇	C₂²×C₄	1	2	yes	—
96	D₄×C₉₇	—	2	1	no	—
97	D₄×C₉₇	C₂²×C₄	1	2	yes	—
98	C₂×C₄×C₉₇	—	2	1	no	—
99	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
100	C₂²⋊D₉₇	—	2	2	no	—
101	C₂²⋊D₉₇	—	2	2	no	—
102	C₂×Dic₉₇	—	2	1	no	—
103	C₂×Dic₉₇	#41	1	1	yes	characteristic
104	D₃₈₈	—	2	1	no	—
105	D₃₈₈	C₂²×C₄	1	2	yes	—
106	C₄×D₉₇	—	2	2	no	—
107	C₄×D₉₇	—	2	2	no	—
108	C₄×D₉₇	C₂²×C₄	1	1	yes	characteristic
109	C₄×D₉₇	C₂×D₄	1	1	yes	characteristic
110	C₄×D₉₇	#41	1	1	yes	characteristic
111	C₄×D₉₇	C₂⁴	1	1	yes	characteristic
112	Dic₁₉₄	—	2	1	no	—
113	Dic₁₉₄	—	2	2	no	—
114	Dic₁₉₄	—	2	2	no	—
115	Dic₁₉₄	C₂²×C₄	1	2	yes	—
116	#11715897	—	2	2	no	—
117	C₈×C₉₇	C₂²×C₄	1	1	yes	characteristic
118	C₈×C₉₇	C₂²×C₄	1	1	yes	characteristic
119	#11715895	C₂²×C₄	1	1	yes	characteristic
120	#11715895	C₂²×C₄	1	1	yes	characteristic
121	C₂²×C₉₇	—	2	2	no	—
122	C₂²×C₉₇	—	8	1	no	—
123	C₂²×C₉₇	C₄×D₄	1	1	yes	characteristic
124	C₂×D₉₇	—	2	2	no	—
125	C₂×D₉₇	—	2	2	no	—
126	C₂×D₉₇	—	4	4	no	—
127	C₂×D₉₇	—	8	2	no	—
128	C₂×D₉₇	—	8	1	no	—
129	C₂×D₉₇	C₄×D₄	1	1	yes	characteristic
130	C₂×D₉₇	C₂²×D₄	1	1	yes	characteristic
131	#87604	—	2	4	no	—
132	#87604	—	4	2	no	—
133	#87604	—	4	2	no	—
134	C₄×C₉₇	—	2	2	no	—
135	C₄×C₉₇	C₄×D₄	1	1	yes	characteristic
136	C₄×C₉₇	C₂³×C₄	1	1	yes	characteristic
137	Dic₉₇	—	2	2	no	—
138	Dic₉₇	C₄×D₄	1	1	yes	characteristic
139	Dic₉₇	C₂×#41	1	1	yes	characteristic
140	C₂×C₉₇	—	2	1	no	—
141	C₂×C₉₇	—	4	2	no	—
142	C₂×C₉₇	C₂×C₄×D₄	1	1	yes	characteristic
143	D₉₇	—	2	1	no	—
144	D₉₇	—	4	2	no	—
145	D₉₇	#577	1	2	yes	—
146	#2949	—	97	1	no	maximal, Sylow
147	C₉₇	#2949	1	1	yes	characteristic, complemented, Sylow
148	C₂²×SD₁₆	—	97	1	no	—
149	#519	—	97	1	no	—
150	#518	—	97	1	no	—
151	#440	—	97	4	no	—
152	C₄×SD₁₆	—	97	4	no	—
153	#427	—	97	1	no	—
154	#419	—	97	1	no	—
155	#418	—	97	1	no	—
156	#405	—	97	1	no	—
157	C₂²×Q₈	—	97	1	no	—
158	C₂²×D₄	—	97	1	no	—
159	C₂×SD₁₆	—	97	1	no	—
160	C₂×SD₁₆	—	97	1	no	—
161	C₂×SD₁₆	—	97	2	no	—
162	C₂×SD₁₆	—	97	2	no	—
163	C₂×SD₁₆	—	97	2	no	—
164	C₂×SD₁₆	—	97	2	no	—
165	C₂×SD₁₆	—	97	1	no	—
166	C₂×SD₁₆	#87604	1	1	yes	characteristic, complemented
167	C₂×M₁₆	—	97	1	no	—
168	C₂²×C₈	—	97	1	no	—
169	C₄×Q₈	—	97	4	no	—
170	C₄×Q₈	—	194	2	no	—
171	C₄×D₄	—	97	4	no	—
172	C₄×D₄	—	194	2	no	—
173	#117	—	97	1	no	—
174	#117	—	97	1	no	—
175	#117	—	97	1	no	—
176	#117	—	194	2	no	—
177	C₂×C₂²⋊C₄	—	194	2	no	—
178	#107	—	97	2	no	—
179	#106	—	97	2	no	—
180	#103	—	97	4	no	—
181	#102	—	97	4	no	—
182	#97	—	97	2	no	—
183	C₄×C₈	—	97	2	no	—
184	C₂⁴	—	194	1	no	—
185	C₂×Q₈	—	97	2	no	—
186	C₂×Q₈	—	97	2	no	—
187	C₂×Q₈	—	97	1	no	—
188	C₂×Q₈	—	194	1	no	—
189	C₂×Q₈	—	194	1	no	—
190	C₂×Q₈	—	194	1	no	—
191	C₂×Q₈	C₂×#87604	1	1	yes	characteristic, complemented
192	C₂×D₄	—	97	2	no	—
193	C₂×D₄	—	97	2	no	—
194	C₂×D₄	—	97	1	no	—
195	C₂×D₄	—	194	1	no	—
196	C₂×D₄	—	194	1	no	—
197	C₂×D₄	—	194	1	no	—
198	C₂×D₄	C₂×#87604	1	1	yes	characteristic
199	C₂²×C₄	—	97	1	no	—
200	C₂²×C₄	—	194	1	no	—
201	C₂²×C₄	—	194	4	no	—
202	SD₁₆	—	97	2	no	—
203	SD₁₆	—	97	2	no	—
204	SD₁₆	—	97	2	no	—
205	SD₁₆	—	97	2	no	—
206	SD₁₆	—	97	2	no	—
207	SD₁₆	—	97	2	no	—
208	SD₁₆	C₂×#87604	1	2	yes	complemented
209	SD₁₆	C₂×#87604	1	2	yes	complemented
210	M₁₆	—	194	1	no	—
211	M₁₆	—	194	1	no	—
212	C₂×C₈	—	97	1	no	—
213	C₂×C₈	—	97	1	no	—
214	C₂×C₈	—	97	1	no	—
215	C₂×C₈	—	97	1	no	—
216	C₂×C₈	—	97	1	no	—
217	C₂×C₈	—	97	2	no	—
218	C₂×C₈	C₂×#87604	1	1	yes	characteristic, complemented
219	C₄⋊C₄	—	97	2	no	—
220	C₄⋊C₄	—	97	2	no	—
221	C₄⋊C₄	—	97	2	no	—
222	C₄⋊C₄	—	194	4	no	—
223	C₂²⋊C₄	—	97	2	no	—
224	C₂²⋊C₄	—	194	4	no	—
225	C₂²⋊C₄	—	194	1	no	—
226	C₂²⋊C₄	—	194	1	no	—
227	C₄²	—	97	2	no	—
228	C₄²	—	194	1	no	—
229	C₄²	—	194	1	no	—
230	C₄²	—	194	4	no	—
231	C₂³	—	2	1	no	—
232	C₂³	—	776	2	no	—
233	C₂³	—	97	1	no	—
234	C₂³	—	194	2	no	—
235	C₂³	—	194	2	no	—
236	C₂³	—	194	1	no	—
237	Q₈	—	2	1	no	—
238	Q₈	—	97	2	no	—
239	Q₈	—	194	2	no	—
240	Q₈	—	194	2	no	—
241	Q₈	—	194	1	no	—
242	Q₈	C₂²×#87604	1	2	yes	—
243	D₄	—	2	1	no	—
244	D₄	—	97	2	no	—
245	D₄	—	194	2	no	—
246	D₄	—	194	2	no	—
247	D₄	—	194	1	no	—
248	D₄	C₂²×#87604	1	2	yes	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	776	4	no	—
251	C₂×C₄	—	97	1	no	—
252	C₂×C₄	—	97	1	no	—
253	C₂×C₄	—	97	1	no	—
254	C₂×C₄	—	97	1	no	—
255	C₂×C₄	—	97	1	no	—
256	C₂×C₄	—	97	4	no	—
257	C₂×C₄	—	194	2	no	—
258	C₂×C₄	—	194	2	no	—
259	C₂×C₄	—	194	2	no	—
260	C₂×C₄	—	194	2	no	—
261	C₂×C₄	—	194	1	no	—
262	C₂×C₄	C₂²×#87604	1	1	yes	characteristic
263	C₈	—	97	1	no	—
264	C₈	—	97	1	no	—
265	C₈	—	194	2	no	—
266	C₈	C₂²×#87604	1	1	yes	characteristic
267	C₈	C₂²×#87604	1	1	yes	characteristic
268	C₂²	—	2	2	no	—
269	C₂²	—	8	1	no	—
270	C₂²	—	776	1	no	—
271	C₂²	—	776	2	no	—
272	C₂²	—	97	1	no	—
273	C₂²	—	97	1	no	—
274	C₂²	—	388	4	no	—
275	C₂²	—	194	2	no	—
276	C₂²	—	194	2	no	—
277	C₂²	D₄×#87604	1	1	yes	characteristic
278	C₄	—	2	2	no	—
279	C₄	—	97	1	no	—
280	C₄	—	97	1	no	—
281	C₄	—	388	2	no	—
282	C₄	—	388	2	no	—
283	C₄	—	194	4	no	—
284	C₄	—	194	2	no	—
285	C₄	D₄×#87604	1	1	yes	characteristic
286	C₄	C₂³×#87604	1	1	yes	characteristic
287	C₂	—	2	1	no	—
288	C₂	—	4	2	no	—
289	C₂	—	97	2	no	—
290	C₂	—	388	2	no	—
291	C₂	—	194	1	no	—
292	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	C₂×SD₁₆×C₉₇	C₄
socle	C₂×C₉₇	C₂×C₄×D₄
prime radical	C₂×D₄×D₉₇	C₂²
cosocle	C₄×D₉₇	C₂⁴
hypercenter	C₂×SD₁₆	#87604
hypocenter	C₉₇	#2949
LCS	G ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×SD₁₆

Complements

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

N	H ≅ G/N	type
1. #20261803	C₂	semidirect
3. #20261774	C₂	semidirect
4. SD₁₆×#87604	C₂	semidirect
5. #20261745	C₂	semidirect
6. #20261728	C₂	semidirect
8. #20261698	C₂	semidirect
12. Q₈×#87604	C₂²	semidirect
20. #15690879	C₂²	semidirect
21. #15690878	C₂²	semidirect
22. #15690873	C₂²	semidirect
24. #15690864	C₂²	semidirect
25. C₈×#87604	C₂²	semidirect
28. C₂×#13095500	C₄	semidirect
29. C₂×#13095499	C₄	semidirect
30. C₂×#13095498	C₄	semidirect
31. SD₁₆×D₉₇	C₄	semidirect
32. SD₁₆×D₉₇	C₄	semidirect
36. C₂×SD₁₆×C₉₇	C₄	semidirect
55. C₂×Dic₁₉₄	C₂×C₄	semidirect
58. Q₈×D₉₇	C₂×C₄	semidirect
68. #13095500	C₂×C₄	semidirect
69. #13095500	C₂×C₄	semidirect
70. #13095499	C₂×C₄	semidirect
71. #13095499	C₂×C₄	semidirect
72. #13095498	C₂×C₄	semidirect
73. #13095498	C₂×C₄	semidirect
74. C₂×#11715895	C₂×C₄	semidirect
75. C₈×D₉₇	C₂×C₄	semidirect
76. C₈×D₉₇	C₂×C₄	semidirect
79. C₂×Q₈×C₉₇	C₂×C₄	semidirect
81. SD₁₆×C₉₇	C₂×C₄	semidirect
82. SD₁₆×C₉₇	C₂×C₄	semidirect
83. C₂×C₈×C₉₇	C₂×C₄	semidirect
147. C₉₇	#2949	semidirect
166. C₂×SD₁₆	#87604	semidirect
191. C₂×Q₈	C₂×#87604	semidirect
208. SD₁₆	C₂×#87604	semidirect
209. SD₁₆	C₂×#87604	semidirect
218. 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
#20261803	2
#20261802	2
#20261774	2
SD₁₆×#87604	2
#20261745	2
#20261728	2
#20261727	2
#20261698	2
C₂×SD₁₆×D₉₇	2
#2949	97