Subgroups — #27158419

Order 12416.

400 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261760	C₂	1	2	yes	maximal, complemented
2	#20261759	C₂	1	2	yes	maximal, complemented
3	#20261754	C₂	1	1	yes	maximal, characteristic, complemented
4	#20261753	C₂	1	1	yes	maximal, characteristic, complemented
5	#20261752	C₂	1	1	yes	maximal, characteristic, complemented
6	#20261751	C₂	1	1	yes	maximal, characteristic, complemented
7	#20261741	C₂	1	2	yes	maximal, complemented
8	#20261708	C₂	1	2	yes	maximal
9	C₄×#13095520	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₄×#11715897	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#13095510	C₂	1	1	yes	maximal, characteristic
12	C₂×#13095520	C₂²	1	2	yes	—
13	C₂²×#11715897	C₂²	1	2	yes	—
14	#15690877	C₂²	1	2	yes	—
15	#15690877	C₂²	1	2	yes	—
16	C₄⋊#11715897	C₂²	1	2	yes	—
17	C₄⋊#11715897	C₂²	1	2	yes	—
18	#15690867	C₂²	1	2	yes	—
19	#15690867	C₂²	1	2	yes	—
20	C₂²⋊#11715897	C₂²	1	2	yes	—
21	C₂²⋊#11715897	C₂²	1	2	yes	—
22	#15690865	C₂²	1	2	yes	—
23	C₄×#11715897	C₂²	1	1	yes	characteristic
24	C₄×#11715897	C₂²	1	1	yes	characteristic
25	C₄×#11715897	C₂²	1	2	yes	—
26	C₄×#11715897	C₂²	1	1	yes	characteristic
27	C₄×#11715897	C₂²	1	1	yes	characteristic
28	C₂×#13095510	C₂²	1	1	yes	characteristic
29	C₄×Dic₁₉₄	C₄	1	1	yes	characteristic
30	#15690775	C₄	1	1	yes	characteristic
31	D₄×Dic₉₇	C₄	1	1	yes	characteristic
32	C₄×C₂²⋊D₉₇	C₄	1	1	yes	characteristic
33	C₄×C₂²⋊D₉₇	C₄	1	1	yes	characteristic
34	#15690770	C₄	1	1	yes	characteristic
35	#15690770	C₄	1	1	yes	characteristic
36	#15690768	C₂²	1	1	yes	characteristic
37	#15690766	C₂²	1	1	yes	characteristic
38	#15690766	C₂²	1	1	yes	characteristic
39	C₂×C₄×Dic₉₇	C₂²	1	1	yes	characteristic
40	C₂×C₄×Dic₉₇	C₂²	1	1	yes	characteristic
41	C₄²×D₉₇	C₂²	1	1	yes	characteristic
42	C₄×D₄×C₉₇	C₄	1	1	yes	characteristic
43	#13095520	—	2	4	no	—
44	C₂×#11715897	—	2	4	no	—
45	C₂×#11715897	C₂³	1	2	yes	—
46	C₂×#11715897	C₂³	1	2	yes	—
47	C₂×#11715897	C₂³	1	2	yes	—
48	C₂×#11715897	C₂³	1	2	yes	—
49	#13095510	—	2	4	no	—
50	C₂×Dic₁₉₄	C₂×C₄	1	1	yes	characteristic
51	C₂×C₂²⋊D₉₇	C₂×C₄	1	1	yes	characteristic
52	C₂×C₂²⋊D₉₇	C₂×C₄	1	1	yes	characteristic
53	C₂²×Dic₉₇	C₂×C₄	1	1	yes	characteristic
54	C₂²×Dic₉₇	C₂×C₄	1	1	yes	characteristic
55	C₂²×Dic₉₇	C₂³	1	1	yes	characteristic
56	C₂²×Dic₉₇	C₂³	1	1	yes	characteristic
57	C₂×C₄×D₉₇	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄×D₉₇	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄×D₉₇	C₂³	1	1	yes	characteristic
60	C₄⋊Dic₉₇	C₂×C₄	1	1	yes	characteristic
61	#13095490	C₂×C₄	1	1	yes	characteristic
62	#13095490	C₂×C₄	1	1	yes	characteristic
63	#13095490	C₂×C₄	1	1	yes	characteristic
64	#13095490	C₂×C₄	1	1	yes	characteristic
65	C₂²⋊Dic₉₇	C₂×C₄	1	1	yes	characteristic
66	C₂²⋊Dic₉₇	C₂×C₄	1	1	yes	characteristic
67	#13095488	C₂×C₄	1	1	yes	characteristic
68	#13095488	C₂×C₄	1	1	yes	characteristic
69	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic
70	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic
71	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic
72	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic
73	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic
74	C₄×Dic₉₇	C₂³	1	1	yes	characteristic
75	C₄×Dic₉₇	C₂³	1	1	yes	characteristic
76	C₄×Dic₉₇	C₂³	1	1	yes	characteristic
77	C₄×Dic₉₇	C₂³	1	1	yes	characteristic
78	C₂×D₄×C₉₇	C₂×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
81	C₄⋊C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
82	C₂²⋊C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
83	C₂²⋊C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
84	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
85	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
86	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
87	C₂²×D₉₇	C₂²×C₄	1	1	yes	characteristic
88	D₄×C₉₇	—	2	2	no	—
89	C₂×C₄×C₉₇	—	2	2	no	—
90	C₂×C₄×C₉₇	—	2	2	no	—
91	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
96	C₂²⋊D₉₇	—	2	2	no	—
97	C₂²⋊D₉₇	—	2	2	no	—
98	C₂×Dic₉₇	—	2	2	no	—
99	C₂×Dic₉₇	—	2	2	no	—
100	C₂×Dic₉₇	—	2	2	no	—
101	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
102	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
103	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
104	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
105	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
106	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
107	C₂×Dic₉₇	#41	1	4	yes	—
108	C₂×Dic₉₇	C₂⁴	1	1	yes	characteristic
109	C₄×D₉₇	—	2	2	no	—
110	C₄×D₉₇	—	2	2	no	—
111	C₄×D₉₇	—	2	2	no	—
112	Dic₁₉₄	—	2	2	no	—
113	#11715897	—	2	2	no	—
114	#11715897	—	2	2	no	—
115	#11715897	—	2	2	no	—
116	#11715897	—	2	2	no	—
117	C₂²×C₉₇	—	2	1	no	—
118	C₂²×C₉₇	—	2	1	no	—
119	C₂²×C₉₇	—	2	1	no	—
120	C₂²×C₉₇	—	2	1	no	—
121	C₂²×C₉₇	—	2	1	no	—
122	C₂²×C₉₇	#117	1	2	yes	—
123	C₂²×C₉₇	C₂³×C₄	1	1	yes	characteristic
124	C₂×D₉₇	—	2	1	no	—
125	C₂×D₉₇	—	2	1	no	—
126	C₂×D₉₇	—	2	1	no	—
127	C₄×C₉₇	—	2	1	no	—
128	C₄×C₉₇	—	2	1	no	—
129	C₄×C₉₇	—	2	1	no	—
130	C₄×C₉₇	—	2	1	no	—
131	C₄×C₉₇	#131	1	2	yes	—
132	Dic₉₇	—	2	1	no	—
133	Dic₉₇	—	2	1	no	—
134	Dic₉₇	—	2	1	no	—
135	Dic₉₇	—	2	1	no	—
136	Dic₉₇	#117	1	2	yes	—
137	Dic₉₇	#131	1	2	yes	—
138	Dic₉₇	C₂×#41	1	2	yes	—
139	C₂×C₉₇	—	2	2	no	—
140	C₂×C₉₇	—	4	1	no	—
141	C₂×C₉₇	C₂×#117	1	1	yes	characteristic
142	C₂×C₉₇	C₂×#131	1	1	yes	characteristic
143	C₂×C₉₇	C₂×#131	1	1	yes	characteristic
144	D₉₇	—	4	1	no	—
145	#2922	—	97	1	no	maximal, Sylow
146	C₉₇	#2922	1	1	yes	characteristic, complemented, Sylow
147	C₄×#41	—	97	1	no	—
148	#433	—	97	2	no	—
149	#433	—	97	2	no	—
150	#431	—	97	1	no	—
151	#431	—	97	1	no	—
152	#431	—	97	1	no	—
153	#431	—	97	1	no	—
154	#424	—	97	2	no	—
155	#408	—	97	2	no	—
156	C₄×M₁₆	—	97	1	no	—
157	C₂×C₄×C₈	—	97	1	no	—
158	C₂×#41	—	97	1	no	—
159	C₂×M₁₆	—	97	2	no	—
160	C₂²×C₈	—	97	2	no	—
161	C₄×Q₈	—	97	1	no	—
162	C₄×Q₈	—	97	1	no	—
163	C₄×D₄	—	97	1	no	—
164	C₄×D₄	—	97	1	no	—
165	C₄×D₄	—	97	1	no	—
166	C₄×D₄	—	97	1	no	—
167	C₄×D₄	—	97	1	no	—
168	C₄×D₄	#87604	1	1	yes	characteristic
169	#117	—	97	1	no	—
170	#117	—	97	1	no	—
171	#117	—	97	1	no	—
172	C₂×C₄²	—	97	1	no	—
173	C₂×C₄²	—	97	1	no	—
174	C₂×C₄²	—	97	1	no	—
175	C₄⋊C₈	—	97	2	no	—
176	C₄⋊C₈	—	97	2	no	—
177	C₄⋊C₈	—	97	2	no	—
178	C₄⋊C₈	—	97	2	no	—
179	C₂²⋊C₈	—	97	2	no	—
180	C₂²⋊C₈	—	97	2	no	—
181	C₂²⋊C₈	—	97	2	no	—
182	C₂²⋊C₈	—	97	2	no	—
183	#97	—	97	2	no	—
184	C₄×C₈	—	97	1	no	—
185	C₄×C₈	—	97	1	no	—
186	C₄×C₈	—	97	2	no	—
187	C₄×C₈	—	97	1	no	—
188	C₄×C₈	—	97	1	no	—
189	#41	—	194	4	no	—
190	C₂×Q₈	—	97	1	no	—
191	C₂×D₄	—	97	1	no	—
192	C₂×D₄	—	97	1	no	—
193	C₂×D₄	C₂×#87604	1	1	yes	characteristic
194	C₂²×C₄	—	97	1	no	—
195	C₂²×C₄	—	97	1	no	—
196	C₂²×C₄	—	97	1	no	—
197	C₂²×C₄	—	97	1	no	—
198	C₂²×C₄	—	97	1	no	—
199	C₂²×C₄	—	97	1	no	—
200	C₂²×C₄	—	97	1	no	—
201	C₂²×C₄	C₂×#87604	1	1	yes	characteristic
202	C₂²×C₄	C₂×#87604	1	1	yes	characteristic
203	M₁₆	—	194	4	no	—
204	C₂×C₈	—	97	2	no	—
205	C₂×C₈	—	97	2	no	—
206	C₂×C₈	—	97	2	no	—
207	C₂×C₈	—	97	2	no	—
208	C₂×C₈	—	194	4	no	—
209	C₄⋊C₄	—	97	1	no	—
210	C₄⋊C₄	—	97	1	no	—
211	C₄⋊C₄	—	97	1	no	—
212	C₄⋊C₄	—	97	1	no	—
213	C₄⋊C₄	—	97	1	no	—
214	C₄⋊C₄	C₂×#87604	1	1	yes	characteristic
215	C₂²⋊C₄	—	97	1	no	—
216	C₂²⋊C₄	—	97	1	no	—
217	C₂²⋊C₄	—	97	1	no	—
218	C₂²⋊C₄	—	97	1	no	—
219	C₂²⋊C₄	C₂×#87604	1	1	yes	characteristic
220	C₂²⋊C₄	C₂×#87604	1	1	yes	characteristic
221	C₄²	—	97	1	no	—
222	C₄²	—	97	1	no	—
223	C₄²	—	97	1	no	—
224	C₄²	—	97	1	no	—
225	C₄²	—	97	1	no	—
226	C₄²	—	97	1	no	—
227	C₄²	—	97	1	no	—
228	C₄²	—	97	1	no	—
229	C₄²	—	97	1	no	—
230	C₄²	C₂×#87604	1	1	yes	characteristic
231	C₂³	—	97	1	no	—
232	C₂³	C₂²×#87604	1	1	yes	characteristic
233	C₂³	C₂²×#87604	1	1	yes	characteristic
234	Q₈	—	194	2	no	—
235	D₄	—	2	2	no	—
236	D₄	—	194	2	no	—
237	D₄	—	194	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	97	4	no	—
241	C₂×C₄	—	97	1	no	—
242	C₂×C₄	—	97	1	no	—
243	C₂×C₄	—	97	1	no	—
244	C₂×C₄	—	97	1	no	—
245	C₂×C₄	—	97	1	no	—
246	C₂×C₄	—	97	1	no	—
247	C₂×C₄	—	97	1	no	—
248	C₂×C₄	—	194	2	no	—
249	C₂×C₄	—	194	2	no	—
250	C₂×C₄	—	194	2	no	—
251	C₂×C₄	—	194	2	no	—
252	C₂×C₄	—	194	2	no	—
253	C₂×C₄	—	194	2	no	—
254	C₂×C₄	C₂²×#87604	1	1	yes	characteristic
255	C₂×C₄	C₂²×#87604	1	1	yes	characteristic
256	C₂×C₄	C₂²×#87604	1	1	yes	characteristic
257	C₂×C₄	C₂²×#87604	1	1	yes	characteristic
258	C₂×C₄	C₂²×#87604	1	1	yes	characteristic
259	C₈	—	194	2	no	—
260	C₈	—	194	2	no	—
261	C₈	—	194	2	no	—
262	C₈	—	194	2	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	2	1	no	—
267	C₂²	—	2	1	no	—
268	C₂²	—	194	1	no	—
269	C₂²	—	194	1	no	—
270	C₂²	—	194	1	no	—
271	C₂²	#15690889	1	2	yes	—
272	C₂²	C₂³×#87604	1	1	yes	characteristic
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	97	2	no	—
278	C₄	—	97	2	no	—
279	C₄	—	97	2	no	—
280	C₄	—	194	1	no	—
281	C₄	—	194	1	no	—
282	C₄	—	194	1	no	—
283	C₄	—	194	1	no	—
284	C₄	#15690897	1	2	yes	—
285	C₂	—	2	2	no	—
286	C₂	—	4	1	no	—
287	C₂	—	388	1	no	—
288	C₂	C₂×#15690889	1	1	yes	characteristic
289	C₂	C₂×#15690897	1	1	yes	characteristic
290	C₂	D₉₇⋊#131	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×#87604
commutator, socle	C₂²×C₉₇	C₂³×C₄
Frattini	C₂²	C₂³×#87604
Fitting	C₄×D₄×C₉₇	C₄
prime radical	C₂×#13095510	C₂²
cosocle	C₂×Dic₉₇	C₂⁴
hypercenter	C₄×D₄	#87604
hypocenter	C₉₇	#2922
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
1. #20261760	C₂	semidirect
2. #20261759	C₂	semidirect
3. #20261754	C₂	semidirect
4. #20261753	C₂	semidirect
5. #20261752	C₂	semidirect
6. #20261751	C₂	semidirect
7. #20261741	C₂	semidirect
9. C₄×#13095520	C₂	semidirect
10. C₂×C₄×#11715897	C₂	semidirect
146. C₉₇	#2922	semidirect

Maximal subgroups

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

subgroup	index
#20261760	2
#20261759	2
#20261754	2
#20261753	2
#20261752	2
#20261751	2
#20261741	2
#20261708	2
C₄×#13095520	2
C₂×C₄×#11715897	2
C₄×#13095510	2
#2922	97