Subgroups — #27150403

Order 12416.

392 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13095511	C₂	1	1	yes	maximal, characteristic
2	#20260797	C₂	1	2	yes	maximal, complemented
3	#20260793	C₂	1	2	yes	maximal, complemented
4	#20260791	C₂	1	1	yes	maximal, characteristic, complemented
5	#20260788	C₂	1	1	yes	maximal, characteristic, complemented
6	#20260787	C₂	1	1	yes	maximal, characteristic, complemented
7	#20260757	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄⋊C₈×D₉₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#20260700	C₂	1	2	yes	maximal
10	#20260692	C₂	1	2	yes	maximal, complemented
11	#432×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13095511	C₂²	1	1	yes	characteristic
13	C₂×#13095494	C₂²	1	2	yes	—
14	C₂×C₈×D₉₇	C₂²	1	2	yes	—
15	C₄×Dic₁₉₄	C₄	1	1	yes	characteristic
16	C₄×Dic₁₉₄	C₄	1	1	yes	characteristic
17	C₄×D₃₈₈	C₄	1	1	yes	characteristic
18	C₄×D₃₈₈	C₄	1	1	yes	characteristic
19	C₄×C₂²⋊D₉₇	C₄	1	1	yes	characteristic
20	C₄×C₂²⋊D₉₇	C₄	1	1	yes	characteristic
21	C₄×C₂²⋊D₉₇	C₄	1	1	yes	characteristic
22	C₄×C₂²⋊D₉₇	C₄	1	1	yes	characteristic
23	#15690769	C₂²	1	1	yes	characteristic
24	#15690767	C₂²	1	1	yes	characteristic
25	#15690767	C₂²	1	1	yes	characteristic
26	C₄²×D₉₇	C₂²	1	1	yes	characteristic
27	C₄²×D₉₇	C₂²	1	1	yes	characteristic
28	C₄⋊#11715895	C₂²	1	1	yes	characteristic
29	C₄⋊#11715895	C₂²	1	1	yes	characteristic
30	#15690739	C₂²	1	2	yes	—
31	#15690739	C₂²	1	2	yes	—
32	C₂²⋊#11715895	C₂²	1	2	yes	—
33	#15690722	C₂²	1	2	yes	—
34	#15690722	C₂²	1	2	yes	—
35	#15690720	C₂²	1	2	yes	—
36	C₄×#11715895	C₂²	1	2	yes	—
37	C₈×Dic₉₇	C₂²	1	2	yes	—
38	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
39	C₄⋊C₈×C₉₇	C₂²	1	1	yes	characteristic
40	C₄⋊C₈×C₉₇	C₂²	1	1	yes	characteristic
41	C₂²⋊C₈×C₉₇	C₂²	1	2	yes	—
42	#97×C₉₇	C₂²	1	2	yes	—
43	#13095511	—	2	2	no	—
44	#13095511	—	2	2	no	—
45	C₂×Dic₁₉₄	C₂×C₄	1	1	yes	characteristic
46	C₂×D₃₈₈	C₂×C₄	1	1	yes	characteristic
47	C₂×C₂²⋊D₉₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₂²⋊D₉₇	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×D₉₇	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×D₉₇	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×D₉₇	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×D₉₇	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄×D₉₇	C₂³	1	1	yes	characteristic
54	C₂×C₄×D₉₇	C₂³	1	1	yes	characteristic
55	#13095494	—	2	4	no	—
56	C₂×#11715895	C₂³	1	2	yes	—
57	C₂×#11715895	C₂³	1	2	yes	—
58	C₈×D₉₇	—	2	4	no	—
59	C₄⋊Dic₉₇	C₂×C₄	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	#13095488	C₂×C₄	1	1	yes	characteristic
70	#13095488	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	C₂×C₄	1	1	yes	characteristic
75	C₄×Dic₉₇	C₂³	1	1	yes	characteristic
76	C₄×Dic₉₇	C₂³	1	1	yes	characteristic
77	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
80	C₂×C₈×C₉₇	C₂³	1	2	yes	—
81	C₂×C₈×C₉₇	C₂³	1	2	yes	—
82	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
83	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
84	C₄²×C₉₇	C₂³	1	1	yes	characteristic
85	C₄²×C₉₇	C₂³	1	1	yes	characteristic
86	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
87	C₂²×D₉₇	C₂²×C₄	1	1	yes	characteristic
88	C₂²×D₉₇	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄×C₉₇	—	2	2	no	—
90	C₂×C₄×C₉₇	—	2	1	no	—
91	C₂×C₄×C₉₇	—	2	1	no	—
92	C₂×C₄×C₉₇	—	4	1	no	—
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	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
99	C₂²⋊D₉₇	—	2	2	no	—
100	C₂²⋊D₉₇	—	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	D₃₈₈	—	2	2	no	—
108	C₄×D₉₇	—	2	2	no	—
109	C₄×D₉₇	—	2	2	no	—
110	C₄×D₉₇	—	2	1	no	—
111	C₄×D₉₇	—	2	1	no	—
112	C₄×D₉₇	—	4	1	no	—
113	C₄×D₉₇	—	4	1	no	—
114	C₄×D₉₇	#41	1	2	yes	—
115	C₄×D₉₇	#41	1	2	yes	—
116	Dic₁₉₄	—	2	2	no	—
117	C₈×C₉₇	—	2	2	no	—
118	C₈×C₉₇	—	2	2	no	—
119	#11715895	—	2	2	no	—
120	#11715895	—	2	2	no	—
121	C₂²×C₉₇	—	2	1	no	—
122	C₂²×C₉₇	—	2	1	no	—
123	C₂²×C₉₇	—	2	1	no	—
124	C₂²×C₉₇	C₂³×C₄	1	1	yes	characteristic
125	C₂×D₉₇	—	2	1	no	—
126	C₂×D₉₇	—	2	1	no	—
127	C₂×D₉₇	—	2	1	no	—
128	C₂×D₉₇	—	2	1	no	—
129	C₂×D₉₇	—	2	1	no	—
130	C₂×D₉₇	#117	1	2	yes	—
131	C₄×C₉₇	—	2	1	no	—
132	C₄×C₉₇	—	2	1	no	—
133	C₄×C₉₇	—	2	1	no	—
134	C₄×C₉₇	—	2	1	no	—
135	C₄×C₉₇	#131	1	2	yes	—
136	C₄×C₉₇	C₂×#41	1	1	yes	characteristic
137	C₄×C₉₇	C₂×#41	1	1	yes	characteristic
138	Dic₉₇	—	2	1	no	—
139	Dic₉₇	—	2	1	no	—
140	Dic₉₇	—	2	1	no	—
141	Dic₉₇	—	2	1	no	—
142	Dic₉₇	—	2	1	no	—
143	Dic₉₇	#117	1	2	yes	—
144	C₂×C₉₇	—	4	1	no	—
145	C₂×C₉₇	C₂×#117	1	1	yes	characteristic
146	C₂×C₉₇	C₂×#131	1	1	yes	characteristic
147	C₂×C₉₇	#568	1	1	yes	characteristic
148	D₉₇	—	2	2	no	—
149	D₉₇	—	4	1	no	—
150	#2925	—	97	1	no	maximal, Sylow
151	C₉₇	#2925	1	1	yes	characteristic, complemented, Sylow
152	C₄×#41	—	97	1	no	—
153	#433	—	97	2	no	—
154	#433	—	97	2	no	—
155	#432	—	97	1	no	—
156	#432	D₉₇	1	1	yes	characteristic, complemented
157	#431	—	97	1	no	—
158	#431	—	97	1	no	—
159	#423	—	97	1	no	—
160	C₂×C₄⋊C₈	—	97	1	no	—
161	#408	—	97	2	no	—
162	#405	—	97	2	no	—
163	C₂×#41	—	97	1	no	—
164	C₂×M₁₆	—	97	2	no	—
165	C₂²×C₈	—	97	2	no	—
166	C₄×Q₈	—	97	1	no	—
167	C₄×Q₈	—	97	1	no	—
168	C₄×D₄	—	97	1	no	—
169	C₄×D₄	—	97	1	no	—
170	C₄×D₄	—	97	1	no	—
171	C₄×D₄	—	97	1	no	—
172	C₄×D₄	—	97	1	no	—
173	C₄×D₄	—	97	1	no	—
174	#117	—	97	1	no	—
175	#117	—	97	1	no	—
176	#117	—	97	1	no	—
177	C₂×C₄²	—	97	1	no	—
178	C₂×C₄²	—	97	1	no	—
179	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
180	C₄⋊C₈	—	97	2	no	—
181	C₄⋊C₈	—	97	2	no	—
182	C₄⋊C₈	—	97	1	no	—
183	C₄⋊C₈	—	97	1	no	—
184	C₄⋊C₈	C₂×D₉₇	1	1	yes	characteristic
185	C₄⋊C₈	C₂×D₉₇	1	1	yes	characteristic
186	C₂²⋊C₈	—	97	2	no	—
187	C₂²⋊C₈	—	97	2	no	—
188	C₂²⋊C₈	—	97	2	no	—
189	C₂²⋊C₈	C₂×D₉₇	1	2	yes	—
190	#97	—	97	2	no	—
191	#97	C₂×D₉₇	1	2	yes	—
192	C₄×C₈	—	97	2	no	—
193	C₄×C₈	—	97	2	no	—
194	#41	—	194	2	no	—
195	#41	—	194	2	no	—
196	C₂×Q₈	—	97	1	no	—
197	C₂×D₄	—	97	1	no	—
198	C₂×D₄	—	97	1	no	—
199	C₂×D₄	—	97	1	no	—
200	C₂²×C₄	—	97	1	no	—
201	C₂²×C₄	—	97	1	no	—
202	C₂²×C₄	—	97	1	no	—
203	C₂²×C₄	—	97	1	no	—
204	C₂²×C₄	—	97	1	no	—
205	C₂²×C₄	—	97	1	no	—
206	C₂²×C₄	C₄×D₉₇	1	1	yes	characteristic
207	C₂²×C₄	C₄×D₉₇	1	1	yes	characteristic
208	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
209	M₁₆	—	194	4	no	—
210	C₂×C₈	—	97	2	no	—
211	C₂×C₈	—	97	2	no	—
212	C₂×C₈	—	194	4	no	—
213	C₂×C₈	C₂²×D₉₇	1	2	yes	—
214	C₂×C₈	C₂²×D₉₇	1	2	yes	—
215	C₄⋊C₄	—	97	1	no	—
216	C₄⋊C₄	—	97	1	no	—
217	C₄⋊C₄	—	97	1	no	—
218	C₄⋊C₄	—	97	1	no	—
219	C₄⋊C₄	—	97	1	no	—
220	C₄⋊C₄	—	97	1	no	—
221	C₂²⋊C₄	—	97	1	no	—
222	C₂²⋊C₄	—	97	1	no	—
223	C₂²⋊C₄	—	97	1	no	—
224	C₂²⋊C₄	—	97	1	no	—
225	C₂²⋊C₄	—	97	1	no	—
226	C₂²⋊C₄	—	97	1	no	—
227	C₄²	—	97	1	no	—
228	C₄²	—	97	1	no	—
229	C₄²	—	97	1	no	—
230	C₄²	—	97	1	no	—
231	C₄²	—	97	1	no	—
232	C₄²	—	97	1	no	—
233	C₄²	C₄×D₉₇	1	1	yes	characteristic
234	C₄²	C₄×D₉₇	1	1	yes	characteristic
235	C₄²	C₂²×D₉₇	1	1	yes	characteristic
236	C₄²	C₂²×D₉₇	1	1	yes	characteristic
237	C₂³	—	97	1	no	—
238	C₂³	—	97	1	no	—
239	C₂³	C₂×C₄×D₉₇	1	1	yes	characteristic
240	Q₈	—	194	2	no	—
241	D₄	—	194	2	no	—
242	D₄	—	194	2	no	—
243	D₄	—	194	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	4	1	no	—
248	C₂×C₄	—	97	1	no	—
249	C₂×C₄	—	97	2	no	—
250	C₂×C₄	—	97	2	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	—	388	1	no	—
257	C₂×C₄	—	388	1	no	—
258	C₂×C₄	—	194	2	no	—
259	C₂×C₄	—	194	2	no	—
260	C₂×C₄	—	194	1	no	—
261	C₂×C₄	—	194	1	no	—
262	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
263	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
264	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
265	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
266	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
267	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
268	C₈	—	2	2	no	—
269	C₈	—	2	2	no	—
270	C₈	—	194	2	no	—
271	C₈	—	194	2	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	97	2	no	—
276	C₂²	—	194	1	no	—
277	C₂²	—	194	1	no	—
278	C₂²	—	194	1	no	—
279	C₂²	—	194	1	no	—
280	C₂²	—	194	1	no	—
281	C₂²	C₂²×C₄×D₉₇	1	1	yes	characteristic
282	C₄	—	2	1	no	—
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	97	2	no	—
287	C₄	—	194	1	no	—
288	C₄	—	194	1	no	—
289	C₄	—	194	1	no	—
290	C₄	—	194	1	no	—
291	C₄	—	194	1	no	—
292	C₄	#15690818	1	2	yes	—
293	C₄	#41×D₉₇	1	1	yes	characteristic
294	C₄	#41×D₉₇	1	1	yes	characteristic
295	C₂	—	4	1	no	—
296	C₂	—	388	1	no	—
297	C₂	—	194	2	no	—
298	C₂	#117×D₉₇	1	1	yes	characteristic
299	C₂	C₂×#15690818	1	1	yes	characteristic
300	C₂	#20261501	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₉₇
commutator, socle	C₂²×C₉₇	C₂³×C₄
Fitting	#432×C₉₇	C₂
prime radical	C₂×#13095511	C₂²
cosocle	C₂×C₄×C₉₇	C₂⁴
hypercenter	#432	D₉₇
hypocenter	C₉₇	#2925
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ #432

Complements

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

N	H ≅ G/N	type
2. #20260797	C₂	semidirect
3. #20260793	C₂	semidirect
4. #20260791	C₂	semidirect
5. #20260788	C₂	semidirect
6. #20260787	C₂	semidirect
7. #20260757	C₂	semidirect
8. C₄⋊C₈×D₉₇	C₂	semidirect
10. #20260692	C₂	semidirect
11. #432×C₉₇	C₂	semidirect
151. C₉₇	#2925	semidirect
156. #432	D₉₇	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13095511	2
#20260797	2
#20260793	2
#20260791	2
#20260788	2
#20260787	2
#20260757	2
C₄⋊C₈×D₉₇	2
#20260700	2
#20260692	2
#432×C₉₇	2
#2925	97