Subgroups — #27151518

Order 12416.

388 conjugacy classes of subgroups.

The subgroup lattice has 270 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	#20261039	C₂	1	2	yes	maximal, complemented
3	#20261032	C₂	1	2	yes	maximal, complemented
4	#20261027	C₂	1	1	yes	maximal, characteristic, complemented
5	#20261022	C₂	1	1	yes	maximal, characteristic, complemented
6	#20261017	C₂	1	1	yes	maximal, characteristic, complemented
7	#20261012	C₂	1	1	yes	maximal, characteristic, complemented
8	#20260771	C₂	1	2	yes	maximal, complemented
9	C₂×C₄⋊#11715895	C₂	1	1	yes	maximal, characteristic, complemented
10	#20260729	C₂	1	2	yes	maximal
11	#533×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13095511	C₂²	1	1	yes	characteristic
13	C₂²×#11715895	C₂²	1	2	yes	—
14	C₄×Dic₁₉₄	C₂²	1	1	yes	characteristic
15	C₄×Dic₁₉₄	C₂²	1	1	yes	characteristic
16	C₄×D₃₈₈	C₂²	1	1	yes	characteristic
17	C₄×D₃₈₈	C₂²	1	1	yes	characteristic
18	C₄×C₂²⋊D₉₇	—	2	1	no	—
19	C₄×C₂²⋊D₉₇	—	2	1	no	—
20	#15690769	C₂²	1	1	yes	characteristic
21	#15690767	—	2	1	no	—
22	C₄²×D₉₇	—	2	1	no	—
23	#15690743	C₂²	1	2	yes	—
24	#15690743	C₂²	1	2	yes	—
25	#15690741	C₂²	1	2	yes	—
26	#15690741	C₂²	1	2	yes	—
27	C₄⋊#11715895	C₂²	1	1	yes	characteristic
28	C₄⋊#11715895	C₂²	1	1	yes	characteristic
29	C₄⋊#11715895	C₂²	1	2	yes	—
30	#15690733	C₂²	1	2	yes	—
31	#15690733	C₂²	1	2	yes	—
32	#15690730	C₂²	1	2	yes	—
33	#15690730	C₂²	1	2	yes	—
34	C₄⋊Q₈×C₉₇	C₂²	1	1	yes	characteristic
35	C₄⋊Q₈×C₉₇	C₂²	1	1	yes	characteristic
36	#125×C₉₇	C₂²	1	2	yes	—
37	C₂²⋊Q₈×C₉₇	—	2	1	no	—
38	C₂²⋊Q₈×C₉₇	—	2	1	no	—
39	C₄×Q₈×C₉₇	—	2	1	no	—
40	C₄×Q₈×C₉₇	—	2	1	no	—
41	#117×C₉₇	C₂²	1	2	yes	—
42	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
43	#13095511	—	2	2	no	—
44	#13095511	Q₈	1	4	yes	complemented
45	C₂×Dic₁₉₄	C₂³	1	1	yes	characteristic
46	C₂×D₃₈₈	C₂³	1	1	yes	characteristic
47	C₂×C₂²⋊D₉₇	—	2	1	no	—
48	C₂×C₄×D₉₇	—	2	1	no	—
49	C₂×C₄×D₉₇	—	2	1	no	—
50	C₂×C₄×D₉₇	—	2	1	no	—
51	C₂×#11715895	—	2	4	no	—
52	C₂×#11715895	C₂³	1	2	yes	—
53	C₂×#11715895	C₂³	1	2	yes	—
54	C₄⋊Dic₉₇	C₂³	1	1	yes	characteristic
55	C₄⋊Dic₉₇	C₂³	1	1	yes	characteristic
56	#13095490	—	2	1	no	—
57	#13095490	—	2	1	no	—
58	C₂²⋊Dic₉₇	—	2	1	no	—
59	#13095488	—	2	1	no	—
60	#13095488	—	2	1	no	—
61	C₄×Dic₉₇	—	2	1	no	—
62	C₄×Dic₉₇	—	2	1	no	—
63	C₄×Dic₉₇	—	2	1	no	—
64	C₂×Q₈×C₉₇	—	2	1	no	—
65	C₂×Q₈×C₉₇	—	2	1	no	—
66	C₂²×C₄×C₉₇	D₄	1	1	yes	characteristic
67	C₂²×C₄×C₉₇	D₄	1	1	yes	characteristic
68	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
69	C₄⋊C₄×C₉₇	—	2	2	no	—
70	C₄⋊C₄×C₉₇	—	2	2	no	—
71	C₄⋊C₄×C₉₇	—	2	1	no	—
72	C₄⋊C₄×C₉₇	—	2	1	no	—
73	C₄⋊C₄×C₉₇	C₂³	1	2	yes	—
74	C₄⋊C₄×C₉₇	C₂³	1	2	yes	—
75	C₂²⋊C₄×C₉₇	—	2	2	no	—
76	C₄²×C₉₇	—	2	2	no	—
77	C₄²×C₉₇	D₄	1	1	yes	characteristic
78	C₄²×C₉₇	D₄	1	1	yes	characteristic
79	C₄²×C₉₇	C₂³	1	1	yes	characteristic
80	C₄²×C₉₇	C₂³	1	1	yes	characteristic
81	C₂³×C₉₇	C₂×D₄	1	1	yes	characteristic
82	C₂²×D₉₇	—	2	1	no	—
83	Q₈×C₉₇	—	4	2	no	—
84	Q₈×C₉₇	—	4	2	no	—
85	C₂×C₄×C₉₇	—	2	2	no	—
86	C₂×C₄×C₉₇	—	2	2	no	—
87	C₂×C₄×C₉₇	—	2	2	no	—
88	C₂×C₄×C₉₇	—	4	1	no	—
89	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
92	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
93	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
94	C₂×C₄×C₉₇	C₂×Q₈	1	2	yes	—
95	C₂×C₄×C₉₇	#41	1	2	yes	—
96	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
97	C₂²⋊D₉₇	—	2	4	no	—
98	C₂×Dic₉₇	—	2	1	no	—
99	C₂×Dic₉₇	—	2	1	no	—
100	C₂×Dic₉₇	—	2	1	no	—
101	D₃₈₈	—	2	1	no	—
102	D₃₈₈	C₂×Q₈	1	2	yes	—
103	C₄×D₉₇	—	2	4	no	—
104	C₄×D₉₇	—	4	2	no	—
105	C₄×D₉₇	—	4	2	no	—
106	Dic₁₉₄	—	2	1	no	—
107	Dic₁₉₄	C₂×Q₈	1	2	yes	—
108	#11715895	—	2	2	no	—
109	#11715895	—	2	2	no	—
110	C₂²×C₉₇	—	2	1	no	—
111	C₂²×C₉₇	—	2	1	no	—
112	C₂²×C₉₇	C₂²⋊Q₈	1	2	yes	—
113	C₂²×C₉₇	C₂²×D₄	1	1	yes	characteristic
114	C₂×D₉₇	—	2	2	no	—
115	C₂×D₉₇	—	4	1	no	—
116	C₂×D₉₇	—	4	1	no	—
117	C₄×C₉₇	—	2	1	no	—
118	C₄×C₉₇	—	2	1	no	—
119	C₄×C₉₇	—	2	1	no	—
120	C₄×C₉₇	—	4	2	no	—
121	C₄×C₉₇	—	4	2	no	—
122	C₄×C₉₇	C₂²⋊Q₈	1	2	yes	—
123	C₄×C₉₇	#135	1	2	yes	—
124	C₄×C₉₇	C₂²×Q₈	1	1	yes	characteristic
125	C₄×C₉₇	C₂×#41	1	1	yes	characteristic
126	Dic₉₇	—	2	2	no	—
127	Dic₉₇	—	4	1	no	—
128	Dic₉₇	—	4	1	no	—
129	C₂×C₉₇	—	2	2	no	—
130	C₂×C₉₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
131	C₂×C₉₇	C₂×#135	1	1	yes	characteristic
132	C₂×C₉₇	#575	1	1	yes	characteristic
133	D₉₇	—	4	2	no	—
134	#3076	—	97	1	no	maximal, Sylow
135	C₉₇	#3076	1	1	yes	characteristic, complemented, Sylow
136	#533	D₉₇	1	1	yes	characteristic, complemented
137	C₄×#41	—	97	1	no	—
138	#479	—	97	2	no	—
139	#478	—	97	2	no	—
140	#477	—	97	1	no	—
141	#476	—	97	1	no	—
142	#475	—	97	1	no	—
143	#474	—	97	1	no	—
144	#427	—	97	2	no	—
145	C₂×C₄⋊C₈	—	97	1	no	—
146	#416	—	97	2	no	—
147	C₂×#41	—	97	1	no	—
148	C₂²×C₈	—	97	2	no	—
149	C₄⋊Q₈	C₂×D₉₇	1	1	yes	characteristic
150	C₄⋊Q₈	C₂×D₉₇	1	1	yes	characteristic
151	#125	C₂×D₉₇	1	2	yes	—
152	C₂²⋊Q₈	—	2	1	no	—
153	C₂²⋊Q₈	—	2	1	no	—
154	C₄×Q₈	—	2	1	no	—
155	C₄×Q₈	—	2	1	no	—
156	C₄×Q₈	—	97	1	no	—
157	C₄×Q₈	—	97	1	no	—
158	C₄×D₄	—	97	1	no	—
159	C₄×D₄	—	97	1	no	—
160	C₄×D₄	—	194	1	no	—
161	C₄×D₄	—	194	1	no	—
162	#117	—	97	1	no	—
163	#117	—	194	1	no	—
164	#117	C₂×D₉₇	1	2	yes	—
165	C₂×C₄²	—	194	1	no	—
166	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
167	#107	—	97	2	no	—
168	#107	—	97	2	no	—
169	#106	—	97	2	no	—
170	#106	—	97	2	no	—
171	C₄⋊C₈	—	97	1	no	—
172	C₄⋊C₈	—	97	1	no	—
173	C₄⋊C₈	—	97	2	no	—
174	#103	—	97	2	no	—
175	#103	—	97	2	no	—
176	#102	—	97	2	no	—
177	#102	—	97	2	no	—
178	#41	—	97	4	no	—
179	#41	—	194	2	no	—
180	C₂×Q₈	—	2	1	no	—
181	C₂×Q₈	—	2	1	no	—
182	C₂×Q₈	—	97	1	no	—
183	C₂×D₄	—	97	1	no	—
184	C₂×D₄	—	194	1	no	—
185	C₂²×C₄	—	194	1	no	—
186	C₂²×C₄	—	194	1	no	—
187	C₂²×C₄	—	194	1	no	—
188	C₂²×C₄	C₂²⋊D₉₇	1	1	yes	characteristic
189	C₂²×C₄	C₂²⋊D₉₇	1	1	yes	characteristic
190	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
191	C₂×C₈	—	97	2	no	—
192	C₂×C₈	—	97	2	no	—
193	C₂×C₈	—	194	4	no	—
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	—	2	2	no	—
196	C₄⋊C₄	—	2	1	no	—
197	C₄⋊C₄	—	2	1	no	—
198	C₄⋊C₄	—	97	1	no	—
199	C₄⋊C₄	—	97	1	no	—
200	C₄⋊C₄	—	194	1	no	—
201	C₄⋊C₄	—	194	1	no	—
202	C₄⋊C₄	C₂²×D₉₇	1	2	yes	—
203	C₄⋊C₄	C₂²×D₉₇	1	2	yes	—
204	C₂²⋊C₄	—	2	2	no	—
205	C₂²⋊C₄	—	194	1	no	—
206	C₂²⋊C₄	—	194	1	no	—
207	C₂²⋊C₄	—	194	1	no	—
208	C₄²	—	2	2	no	—
209	C₄²	—	194	1	no	—
210	C₄²	—	194	1	no	—
211	C₄²	—	194	1	no	—
212	C₄²	C₂²⋊D₉₇	1	1	yes	characteristic
213	C₄²	C₂²⋊D₉₇	1	1	yes	characteristic
214	C₄²	C₂²×D₉₇	1	1	yes	characteristic
215	C₄²	C₂²×D₉₇	1	1	yes	characteristic
216	C₂³	—	194	1	no	—
217	C₂³	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
218	Q₈	—	4	2	no	—
219	Q₈	—	4	2	no	—
220	Q₈	—	97	2	no	—
221	Q₈	—	194	1	no	—
222	D₄	—	97	2	no	—
223	D₄	—	194	4	no	—
224	D₄	—	194	1	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	4	1	no	—
229	C₂×C₄	—	388	2	no	—
230	C₂×C₄	—	388	2	no	—
231	C₂×C₄	—	194	1	no	—
232	C₂×C₄	—	194	1	no	—
233	C₂×C₄	—	194	1	no	—
234	C₂×C₄	—	194	4	no	—
235	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
236	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
237	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
238	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
239	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
240	C₂×C₄	Q₈×D₉₇	1	2	yes	—
241	C₂×C₄	#13095512	1	2	yes	—
242	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
243	C₈	—	194	2	no	—
244	C₈	—	194	2	no	—
245	C₂²	—	2	1	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	388	1	no	—
248	C₂²	—	388	1	no	—
249	C₂²	—	194	2	no	—
250	C₂²	#15690786	1	2	yes	—
251	C₂²	C₂²×C₂²⋊D₉₇	1	1	yes	characteristic
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	2	1	no	—
255	C₄	—	4	2	no	—
256	C₄	—	4	2	no	—
257	C₄	—	388	1	no	—
258	C₄	—	388	1	no	—
259	C₄	—	194	2	no	—
260	C₄	#15690786	1	2	yes	—
261	C₄	#15690830	1	2	yes	—
262	C₄	C₂×Q₈×D₉₇	1	1	yes	characteristic
263	C₄	C₂×#13095512	1	1	yes	characteristic
264	C₂	—	2	2	no	—
265	C₂	—	388	2	no	—
266	C₂	C₂×#15690786	1	1	yes	characteristic
267	C₂	C₂×#15690830	1	1	yes	characteristic
268	C₂	#20261533	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₉₇
commutator, cosocle	C₂×C₄×C₉₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₉₇
Fitting	#533×C₉₇	C₂
socle	C₂²×C₉₇	C₂²×D₄
prime radical	C₂×#13095511	C₂²
hypercenter	#533	D₉₇
hypocenter	C₉₇	#3076
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

Complements

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

N	H ≅ G/N	type
2. #20261039	C₂	semidirect
3. #20261032	C₂	semidirect
4. #20261027	C₂	semidirect
5. #20261022	C₂	semidirect
6. #20261017	C₂	semidirect
7. #20261012	C₂	semidirect
8. #20260771	C₂	semidirect
9. C₂×C₄⋊#11715895	C₂	semidirect
11. #533×C₉₇	C₂	semidirect
44. #13095511	Q₈	semidirect
135. C₉₇	#3076	semidirect
136. #533	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
#20261039	2
#20261032	2
#20261027	2
#20261022	2
#20261017	2
#20261012	2
#20260771	2
C₂×C₄⋊#11715895	2
#20260729	2
#533×C₉₇	2
#3076	97