Subgroups — #27150769

Order 12416.

464 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×D₄×D₉₇	C₂	1	1	yes	maximal, characteristic
2	#20260814	C₂	1	1	yes	maximal, characteristic, complemented
3	#20260812	C₂	1	1	yes	maximal, characteristic, complemented
4	#20260806	C₂	1	2	yes	maximal, complemented
5	#20260803	C₂	1	2	yes	maximal, complemented
6	D₄×#11715895	C₂	1	1	yes	maximal, characteristic, complemented
7	#20260791	C₂	1	1	yes	maximal, characteristic, complemented
8	#20260786	C₂	1	1	yes	maximal, characteristic, complemented
9	#20260697	C₂	1	2	yes	maximal, complemented
10	C₂×#15690722	C₂	1	2	yes	maximal, complemented
11	C₈×D₄×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₄×D₉₇	C₄	1	1	yes	characteristic
13	C₂²×C₄×D₉₇	C₂²	1	2	yes	—
14	C₂×#13095494	C₂²	1	2	yes	complemented
15	C₂×#13095494	C₂²	1	2	yes	complemented
16	C₂²×#11715895	C₂²	1	2	yes	complemented
17	C₄×D₃₈₈	C₂²	1	1	yes	characteristic
18	D₄×Dic₉₇	C₄	1	1	yes	characteristic
19	#15690772	C₄	1	1	yes	characteristic
20	C₄×C₂²⋊D₉₇	C₂²	1	2	yes	—
21	#15690770	C₄	1	2	yes	—
22	C₄⋊C₄×D₉₇	C₄	1	1	yes	characteristic
23	C₂²⋊C₄×D₉₇	C₄	1	2	yes	—
24	C₄²×D₉₇	C₂²	1	1	yes	characteristic
25	C₄⋊#11715895	C₂²	1	1	yes	characteristic, complemented
26	#15690739	C₂²	1	1	yes	characteristic, complemented
27	#15690739	C₂²	1	1	yes	characteristic, complemented
28	C₂²⋊#11715895	C₂²	1	2	yes	complemented
29	#15690722	C₂²	1	1	yes	characteristic
30	#15690722	C₂²	1	2	yes	—
31	#15690722	C₂²	1	1	yes	characteristic
32	#15690722	C₂²	1	2	yes	complemented
33	#15690722	C₂²	1	2	yes	complemented
34	#15690720	C₂²	1	2	yes	complemented
35	C₄×#11715895	C₂²	1	1	yes	characteristic, complemented
36	C₂²×C₈×C₉₇	C₂²	1	2	yes	complemented
37	C₄×D₄×C₉₇	C₂²	1	1	yes	characteristic
38	C₄⋊C₈×C₉₇	C₂²	1	1	yes	characteristic, complemented
39	C₂²⋊C₈×C₉₇	C₂²	1	2	yes	complemented
40	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic, complemented
41	C₂³×D₉₇	C₂×C₄	1	2	yes	—
42	C₂×D₃₈₈	C₂×C₄	1	1	yes	characteristic
43	C₂×C₂²⋊D₉₇	C₂×C₄	1	2	yes	—
44	D₄×D₉₇	—	2	4	no	—
45	C₂²×Dic₉₇	C₂×C₄	1	2	yes	—
46	C₂×C₄×D₉₇	—	4	2	no	—
47	C₂×C₄×D₉₇	—	4	2	no	—
48	C₂×C₄×D₉₇	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×D₉₇	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×D₉₇	C₂×C₄	1	2	yes	—
51	C₂×C₄×D₉₇	C₂×C₄	1	2	yes	—
52	C₂×C₄×D₉₇	C₂³	1	1	yes	characteristic
53	C₂×C₄×D₉₇	C₂³	1	2	yes	—
54	#13095494	—	4	2	no	—
55	#13095494	—	4	2	no	—
56	C₂×#11715895	—	4	2	no	—
57	C₂×#11715895	C₂³	1	1	yes	characteristic
58	C₂×#11715895	C₂³	1	2	yes	—
59	C₂×#11715895	C₂³	1	1	yes	characteristic
60	C₄⋊Dic₉₇	C₂×C₄	1	1	yes	characteristic
61	#13095490	C₂×C₄	1	2	yes	—
62	C₂²⋊Dic₉₇	C₂×C₄	1	2	yes	—
63	#13095488	C₂×C₄	1	2	yes	—
64	#13095488	C₂×C₄	1	2	yes	—
65	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic
66	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic
67	C₄×Dic₉₇	C₂³	1	1	yes	characteristic
68	C₂×D₄×C₉₇	C₂×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₉₇	C₂³	1	2	yes	—
70	C₂×C₈×C₉₇	—	4	2	no	—
71	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
72	C₂×C₈×C₉₇	C₂³	1	2	yes	—
73	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
74	C₄⋊C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
75	C₂²⋊C₄×C₉₇	C₂×C₄	1	2	yes	—
76	C₄²×C₉₇	C₂³	1	1	yes	characteristic
77	C₂³×C₉₇	C₂²×C₄	1	2	yes	—
78	C₂²×D₉₇	—	2	4	no	—
79	C₂²×D₉₇	—	2	4	no	—
80	C₂²×D₉₇	—	4	2	no	—
81	C₂²×D₉₇	C₂²×C₄	1	2	yes	—
82	C₂²×D₉₇	C₂²×C₄	1	1	yes	characteristic
83	D₄×C₉₇	M₁₆	1	4	yes	complemented
84	C₂×C₄×C₉₇	—	2	2	no	—
85	C₂×C₄×C₉₇	—	2	2	no	—
86	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
88	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
90	C₂²⋊D₉₇	—	2	4	no	—
91	C₂×Dic₉₇	—	4	2	no	—
92	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
93	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
94	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
95	C₂×Dic₉₇	C₂²×C₄	1	2	yes	—
96	D₃₈₈	—	2	2	no	—
97	C₄×D₉₇	—	2	2	no	—
98	C₄×D₉₇	—	2	2	no	—
99	C₄×D₉₇	—	2	2	no	—
100	C₄×D₉₇	—	2	2	no	—
101	C₄×D₉₇	—	2	1	no	—
102	C₄×D₉₇	—	2	1	no	—
103	C₄×D₉₇	—	4	2	no	—
104	C₄×D₉₇	—	4	2	no	—
105	C₈×C₉₇	—	2	1	no	—
106	C₈×C₉₇	—	2	2	no	—
107	C₈×C₉₇	—	2	1	no	—
108	#11715895	—	2	1	no	—
109	#11715895	—	2	2	no	—
110	#11715895	—	2	1	no	—
111	C₂²×C₉₇	—	2	2	no	—
112	C₂²×C₉₇	—	2	2	no	—
113	C₂²×C₉₇	C₂×M₁₆	1	4	yes	—
114	C₂²×C₉₇	C₂³×C₄	1	1	yes	characteristic
115	C₂×D₉₇	—	2	1	no	—
116	C₂×D₉₇	—	2	1	no	—
117	C₂×D₉₇	—	2	2	no	—
118	C₂×D₉₇	—	2	2	no	—
119	C₂×D₉₇	—	2	2	no	—
120	C₂×D₉₇	—	4	8	no	—
121	C₂×D₉₇	#131	1	2	yes	—
122	C₄×C₉₇	—	2	2	no	—
123	C₄×C₉₇	—	2	1	no	—
124	C₄×C₉₇	C₂×M₁₆	1	2	yes	—
125	C₄×C₉₇	H(5,𝔽₂)	1	1	yes	characteristic
126	C₄×C₉₇	H(5,𝔽₂)	1	1	yes	characteristic
127	Dic₉₇	—	2	1	no	—
128	Dic₉₇	—	2	2	no	—
129	Dic₉₇	—	2	1	no	—
130	Dic₉₇	—	2	1	no	—
131	C₂×C₉₇	—	2	4	no	—
132	C₂×C₉₇	#518	1	1	yes	characteristic
133	C₂×C₉₇	C₂²×M₁₆	1	1	yes	characteristic
134	C₂×C₉₇	C₂×#131	1	1	yes	characteristic
135	D₉₇	—	2	2	no	—
136	D₉₇	—	4	2	no	—
137	#2974	—	97	1	no	maximal, Sylow
138	C₉₇	#2974	1	1	yes	characteristic, complemented, Sylow
139	C₂×C₄×D₄	—	97	1	no	—
140	#436	—	97	1	no	—
141	#436	—	97	1	no	—
142	#435	—	97	2	no	—
143	#435	—	97	2	no	—
144	C₈×D₄	—	97	1	no	—
145	C₈×D₄	D₉₇	1	1	yes	characteristic, complemented
146	#432	—	97	1	no	—
147	#431	—	97	1	no	—
148	#407	—	97	2	no	—
149	C₂×C₂²⋊C₈	—	97	2	no	—
150	C₂²×D₄	—	97	1	no	—
151	C₂³×C₄	—	97	2	no	—
152	C₂×M₁₆	—	97	2	no	—
153	C₂×M₁₆	—	97	2	no	—
154	C₂²×C₈	—	97	2	no	—
155	C₂²×C₈	C₂×D₉₇	1	2	yes	complemented
156	C₄×D₄	—	97	1	no	—
157	C₄×D₄	—	97	2	no	—
158	C₄×D₄	—	97	1	no	—
159	C₄×D₄	—	97	2	no	—
160	C₄×D₄	—	97	1	no	—
161	C₄×D₄	C₂×D₉₇	1	1	yes	characteristic
162	C₂×C₄⋊C₄	—	97	1	no	—
163	C₂×C₂²⋊C₄	—	97	2	no	—
164	C₂×C₄²	—	97	1	no	—
165	C₄⋊C₈	—	97	1	no	—
166	C₄⋊C₈	—	97	1	no	—
167	C₄⋊C₈	—	97	1	no	—
168	C₄⋊C₈	C₂×D₉₇	1	1	yes	characteristic, complemented
169	C₂²⋊C₈	—	97	2	no	—
170	C₂²⋊C₈	—	97	1	no	—
171	C₂²⋊C₈	—	97	2	no	—
172	C₂²⋊C₈	—	97	2	no	—
173	C₂²⋊C₈	—	97	2	no	—
174	C₂²⋊C₈	—	97	1	no	—
175	C₂²⋊C₈	C₂×D₉₇	1	2	yes	complemented
176	#97	—	97	2	no	—
177	C₄×C₈	—	97	1	no	—
178	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic, complemented
179	C₂⁴	—	97	2	no	—
180	C₂×D₄	—	97	2	no	—
181	C₂×D₄	—	97	1	no	—
182	C₂×D₄	—	194	4	no	—
183	C₂×D₄	C₄×D₉₇	1	1	yes	characteristic
184	C₂²×C₄	—	97	2	no	—
185	C₂²×C₄	—	97	2	no	—
186	C₂²×C₄	—	97	2	no	—
187	C₂²×C₄	—	97	1	no	—
188	C₂²×C₄	—	97	1	no	—
189	C₂²×C₄	—	97	1	no	—
190	C₂²×C₄	—	97	2	no	—
191	C₂²×C₄	—	388	2	no	—
192	C₂²×C₄	—	388	2	no	—
193	C₂²×C₄	C₂²×D₉₇	1	2	yes	—
194	M₁₆	—	388	2	no	—
195	M₁₆	—	388	2	no	—
196	C₂×C₈	—	4	2	no	—
197	C₂×C₈	—	97	1	no	—
198	C₂×C₈	—	97	1	no	—
199	C₂×C₈	—	97	2	no	—
200	C₂×C₈	—	388	2	no	—
201	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
202	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
203	C₂×C₈	C₂²×D₉₇	1	2	yes	—
204	C₄⋊C₄	—	97	2	no	—
205	C₄⋊C₄	—	97	1	no	—
206	C₄⋊C₄	C₄×D₉₇	1	1	yes	characteristic
207	C₂²⋊C₄	—	97	2	no	—
208	C₂²⋊C₄	—	97	2	no	—
209	C₂²⋊C₄	—	97	2	no	—
210	C₂²⋊C₄	C₄×D₉₇	1	2	yes	—
211	C₄²	—	97	1	no	—
212	C₄²	—	97	1	no	—
213	C₄²	—	97	1	no	—
214	C₄²	C₂²×D₉₇	1	1	yes	characteristic
215	C₂³	—	97	1	no	—
216	C₂³	—	97	2	no	—
217	C₂³	—	388	2	no	—
218	C₂³	—	194	4	no	—
219	C₂³	—	194	4	no	—
220	C₂³	C₂×C₄×D₉₇	1	2	yes	—
221	D₄	—	194	4	no	—
222	D₄	—	194	2	no	—
223	D₄	#13095494	1	4	yes	complemented
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	97	1	no	—
227	C₂×C₄	—	97	1	no	—
228	C₂×C₄	—	97	1	no	—
229	C₂×C₄	—	97	2	no	—
230	C₂×C₄	—	388	2	no	—
231	C₂×C₄	—	388	2	no	—
232	C₂×C₄	—	388	2	no	—
233	C₂×C₄	—	194	2	no	—
234	C₂×C₄	—	194	2	no	—
235	C₂×C₄	—	194	2	no	—
236	C₂×C₄	—	194	1	no	—
237	C₂×C₄	—	194	2	no	—
238	C₂×C₄	—	194	1	no	—
239	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
240	C₂×C₄	C₂×C₄×D₉₇	1	2	yes	—
241	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
242	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
243	C₈	—	2	1	no	—
244	C₈	—	2	2	no	—
245	C₈	—	2	1	no	—
246	C₈	—	194	1	no	—
247	C₈	—	194	1	no	—
248	C₈	—	194	2	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	97	2	no	—
252	C₂²	—	388	8	no	—
253	C₂²	—	194	2	no	—
254	C₂²	—	194	2	no	—
255	C₂²	—	194	2	no	—
256	C₂²	—	194	1	no	—
257	C₂²	—	194	1	no	—
258	C₂²	C₂×#13095494	1	4	yes	—
259	C₂²	C₂²×C₄×D₉₇	1	1	yes	characteristic
260	C₄	—	2	2	no	—
261	C₄	—	2	1	no	—
262	C₄	—	194	1	no	—
263	C₄	—	194	2	no	—
264	C₄	—	194	1	no	—
265	C₄	—	194	1	no	—
266	C₄	C₂×#13095494	1	2	yes	—
267	C₄	#15690855	1	1	yes	characteristic
268	C₄	#15690856	1	1	yes	characteristic
269	C₂	—	2	4	no	—
270	C₂	—	388	2	no	—
271	C₂	—	194	2	no	—
272	C₂	#20261207	1	1	yes	characteristic
273	C₂	C₂²×#13095494	1	1	yes	characteristic
274	C₂	#131×D₉₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₉₇
commutator, socle	C₂²×C₉₇	C₂³×C₄
Fitting	C₈×D₄×C₉₇	C₂
prime radical	C₂×D₄×D₉₇	C₄
cosocle	C₂×C₄×C₉₇	C₂⁴
hypercenter	C₈×D₄	D₉₇
hypocenter	C₉₇	#2974
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₈×D₄

Complements

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

N	H ≅ G/N	type
2. #20260814	C₂	semidirect
3. #20260812	C₂	semidirect
4. #20260806	C₂	semidirect
5. #20260803	C₂	semidirect
6. D₄×#11715895	C₂	semidirect
7. #20260791	C₂	semidirect
8. #20260786	C₂	semidirect
9. #20260697	C₂	semidirect
10. C₂×#15690722	C₂	semidirect
11. C₈×D₄×C₉₇	C₂	semidirect
14. C₂×#13095494	C₂²	semidirect
15. C₂×#13095494	C₂²	semidirect
16. C₂²×#11715895	C₂²	semidirect
25. C₄⋊#11715895	C₂²	semidirect
26. #15690739	C₂²	semidirect
27. #15690739	C₂²	semidirect
28. C₂²⋊#11715895	C₂²	semidirect
32. #15690722	C₂²	semidirect
33. #15690722	C₂²	semidirect
34. #15690720	C₂²	semidirect
35. C₄×#11715895	C₂²	semidirect
36. C₂²×C₈×C₉₇	C₂²	semidirect
38. C₄⋊C₈×C₉₇	C₂²	semidirect
39. C₂²⋊C₈×C₉₇	C₂²	semidirect
40. C₄×C₈×C₉₇	C₂²	semidirect
83. D₄×C₉₇	M₁₆	semidirect
138. C₉₇	#2974	semidirect
145. C₈×D₄	D₉₇	semidirect
155. C₂²×C₈	C₂×D₉₇	semidirect
168. C₄⋊C₈	C₂×D₉₇	semidirect
175. C₂²⋊C₈	C₂×D₉₇	semidirect
178. C₄×C₈	C₂×D₉₇	semidirect
223. D₄	#13095494	semidirect

Maximal subgroups

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

subgroup	index
C₄×D₄×D₉₇	2
#20260814	2
#20260812	2
#20260806	2
#20260803	2
D₄×#11715895	2
#20260791	2
#20260786	2
#20260697	2
C₂×#15690722	2
C₈×D₄×C₉₇	2
#2974	97