Subgroups — C₂×#27161738 #38756178

Order 25216.

1620 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#15691358	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×Dic₃₉₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#27161738	C₂	1	16	yes	maximal, direct product factor
4	C₂×#20263314	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#20263312	C₂	1	2	yes	maximal, complemented
6	C₂×#20263305	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#20263304	C₂	1	2	yes	maximal, complemented
8	C₂×#20263300	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#20263298	C₂	1	2	yes	maximal, complemented
10	C₂×C₄×Dic₃₉₄	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#20263286	C₂	1	2	yes	maximal, complemented
12	C₂×C₄×D₄×C₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#15691358	—	2	8	no	—
14	C₂×#15691358	C₂²	1	8	yes	complemented
15	C₂²×Dic₃₉₄	—	2	8	no	—
16	C₂²×Dic₃₉₄	C₂²	1	2	yes	—
17	C₂²×Dic₃₉₄	C₂²	1	1	yes	characteristic
18	C₂²×Dic₃₉₄	C₂²	1	1	yes	characteristic
19	C₂²×Dic₃₉₄	C₂²	1	8	yes	complemented
20	C₂²×D₇₈₈	C₂²	1	1	yes	characteristic
21	C₂²×C₂²⋊D₁₉₇	C₂²	1	2	yes	—
22	C₂²×C₂²⋊D₁₉₇	C₂²	1	2	yes	—
23	C₂³×Dic₁₉₇	C₂²	1	2	yes	—
24	C₂²×C₄×D₁₉₇	C₂²	1	2	yes	—
25	#20263314	C₂²	1	8	yes	complemented
26	#20263312	C₂²	1	16	yes	complemented
27	#20263305	C₂²	1	8	yes	complemented
28	#20263304	C₂²	1	16	yes	complemented
29	#20263300	C₂²	1	8	yes	complemented
30	#20263298	C₂²	1	16	yes	complemented
31	C₄×Dic₃₉₄	C₂²	1	8	yes	complemented
32	#20263286	C₂²	1	16	yes	complemented
33	C₂×C₄⋊Dic₁₉₇	C₂²	1	1	yes	characteristic
34	C₂×#15691337	C₂²	1	2	yes	—
35	C₂×#15691337	C₂²	1	2	yes	—
36	C₂×C₂²⋊Dic₁₉₇	C₂²	1	2	yes	—
37	C₂×#15691335	C₂²	1	2	yes	—
38	C₂×#15691335	C₂²	1	2	yes	—
39	C₂×#15691335	C₂²	1	2	yes	—
40	C₂×C₄×Dic₁₉₇	C₂²	1	2	yes	—
41	C₂²×D₄×C₁₉₇	C₂²	1	1	yes	characteristic
42	C₂³×C₄×C₁₉₇	C₂²	1	1	yes	characteristic
43	C₂³×C₄×C₁₉₇	C₂²	1	1	yes	characteristic
44	C₄×D₄×C₁₉₇	C₂²	1	8	yes	complemented
45	C₂×C₄⋊C₄×C₁₉₇	C₂²	1	1	yes	characteristic
46	C₂×C₂²⋊C₄×C₁₉₇	C₂²	1	1	yes	characteristic
47	C₂×C₂²⋊C₄×C₁₉₇	C₂²	1	1	yes	characteristic
48	C₂×C₄²×C₁₉₇	C₂²	1	1	yes	characteristic
49	C₂³×D₁₉₇	C₂³	1	2	yes	—
50	#15691358	—	2	32	no	—
51	C₂×Dic₃₉₄	—	2	32	no	—
52	C₂×Dic₃₉₄	—	2	8	no	—
53	C₂×Dic₃₉₄	—	2	4	no	—
54	C₂×Dic₃₉₄	D₄	1	8	yes	complemented
55	C₂×Dic₃₉₄	C₂³	1	8	yes	—
56	C₂×Dic₃₉₄	C₂³	1	4	yes	—
57	C₂×Dic₃₉₄	C₂³	1	4	yes	—
58	C₂×D₇₈₈	—	2	4	no	—
59	C₂×D₇₈₈	C₂³	1	4	yes	—
60	C₂×C₂²⋊D₁₉₇	—	2	8	no	—
61	C₂×C₂²⋊D₁₉₇	—	2	8	no	—
62	C₂×C₂²⋊D₁₉₇	C₂³	1	8	yes	—
63	C₂×C₂²⋊D₁₉₇	C₂³	1	8	yes	—
64	C₂²×Dic₁₉₇	—	2	8	no	—
65	C₂²×Dic₁₉₇	C₂³	1	2	yes	—
66	C₂²×Dic₁₉₇	C₂³	1	2	yes	—
67	C₂²×Dic₁₉₇	C₂³	1	8	yes	—
68	C₂²×Dic₁₉₇	C₂³	1	2	yes	—
69	C₂×C₄×D₁₉₇	—	2	8	no	—
70	C₂×C₄×D₁₉₇	C₂³	1	8	yes	—
71	C₄⋊Dic₁₉₇	C₂³	1	4	yes	—
72	#15691337	C₂³	1	8	yes	—
73	#15691337	C₂³	1	8	yes	—
74	C₂²⋊Dic₁₉₇	C₂³	1	8	yes	—
75	#15691335	C₂³	1	8	yes	—
76	#15691335	C₂³	1	8	yes	—
77	#15691335	C₂³	1	8	yes	—
78	C₄×Dic₁₉₇	C₂³	1	8	yes	—
79	C₂⁴×C₁₉₇	C₂³	1	1	yes	characteristic
80	C₂⁴×C₁₉₇	C₂³	1	1	yes	characteristic
81	C₂×D₄×C₁₉₇	—	2	4	no	—
82	C₂×D₄×C₁₉₇	C₂³	1	4	yes	—
83	C₂²×C₄×C₁₉₇	—	2	4	no	—
84	C₂²×C₄×C₁₉₇	—	2	4	no	—
85	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
86	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
87	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
88	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
89	C₂²×C₄×C₁₉₇	C₂³	1	4	yes	—
90	C₂²×C₄×C₁₉₇	C₂³	1	4	yes	—
91	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
92	C₄⋊C₄×C₁₉₇	C₂³	1	4	yes	—
93	C₂²⋊C₄×C₁₉₇	C₂³	1	4	yes	—
94	C₂²⋊C₄×C₁₉₇	C₂³	1	4	yes	—
95	C₄²×C₁₉₇	C₂³	1	4	yes	—
96	C₂²×D₁₉₇	—	2	4	no	—
97	C₂²×D₁₉₇	—	2	4	no	—
98	C₂²×D₁₉₇	—	2	4	no	—
99	C₂²×D₁₉₇	C₂⁴	1	4	yes	—
100	Dic₃₉₄	—	2	8	no	—
101	Dic₃₉₄	—	2	16	no	—
102	Dic₃₉₄	C₂×D₄	1	16	yes	complemented
103	D₇₈₈	—	2	8	no	—
104	C₂²⋊D₁₉₇	—	2	16	no	—
105	C₂²⋊D₁₉₇	—	2	16	no	—
106	C₂×Dic₁₉₇	—	2	4	no	—
107	C₂×Dic₁₉₇	—	2	4	no	—
108	C₂×Dic₁₉₇	—	2	16	no	—
109	C₂×Dic₁₉₇	C₂×D₄	1	8	yes	—
110	C₂×Dic₁₉₇	C₂⁴	1	4	yes	—
111	C₂×Dic₁₉₇	C₂⁴	1	4	yes	—
112	C₂×Dic₁₉₇	C₂⁴	1	4	yes	—
113	C₄×D₁₉₇	—	2	16	no	—
114	C₂³×C₁₉₇	—	2	2	no	—
115	C₂³×C₁₉₇	—	2	2	no	—
116	C₂³×C₁₉₇	—	2	2	no	—
117	C₂³×C₁₉₇	—	2	2	no	—
118	C₂³×C₁₉₇	—	2	2	no	—
119	C₂³×C₁₉₇	#41	1	4	yes	complemented
120	C₂³×C₁₉₇	C₂⁴	1	2	yes	—
121	C₂³×C₁₉₇	C₂⁴	1	2	yes	—
122	C₂³×C₁₉₇	C₂⁴	1	1	yes	characteristic
123	D₄×C₁₉₇	—	2	8	no	—
124	C₂×C₄×C₁₉₇	—	2	2	no	—
125	C₂×C₄×C₁₉₇	—	2	2	no	—
126	C₂×C₄×C₁₉₇	—	2	2	no	—
127	C₂×C₄×C₁₉₇	—	2	2	no	—
128	C₂×C₄×C₁₉₇	—	2	8	no	—
129	C₂×C₄×C₁₉₇	—	2	8	no	—
130	C₂×C₄×C₁₉₇	C₂×D₄	1	4	yes	—
131	C₂×C₄×C₁₉₇	C₂⁴	1	2	yes	—
132	C₂×C₄×C₁₉₇	C₂⁴	1	2	yes	—
133	C₂×C₄×C₁₉₇	C₂⁴	1	2	yes	—
134	C₂×C₄×C₁₉₇	C₂⁴	1	2	yes	—
135	C₂×C₄×C₁₉₇	C₂⁴	1	2	yes	—
136	C₂²×C₁₉₇	—	2	8	no	—
137	C₂²×C₁₉₇	—	2	2	no	—
138	C₂²×C₁₉₇	—	2	2	no	—
139	C₂²×C₁₉₇	—	2	2	no	—
140	C₂²×C₁₉₇	—	2	2	no	—
141	C₂²×C₁₉₇	—	2	2	no	—
142	C₂²×C₁₉₇	—	4	4	no	—
143	C₂²×C₁₉₇	C₂²×D₄	1	2	yes	—
144	C₂²×C₁₉₇	C₂×#41	1	2	yes	—
145	C₂²×C₁₉₇	C₂×#41	1	4	yes	complemented
146	C₂²×C₁₉₇	2₋¹⁺⁴	1	2	yes	—
147	C₂²×C₁₉₇	C₂⁵	1	1	yes	characteristic
148	C₂×D₁₉₇	—	2	4	no	—
149	C₂×D₁₉₇	—	2	4	no	—
150	C₂×D₁₉₇	—	2	4	no	—
151	C₂×D₁₉₇	—	4	8	no	—
152	C₄×C₁₉₇	—	2	2	no	—
153	C₄×C₁₉₇	—	2	2	no	—
154	C₄×C₁₉₇	—	2	2	no	—
155	C₄×C₁₉₇	—	2	2	no	—
156	C₄×C₁₉₇	C₂²×D₄	1	4	yes	—
157	Dic₁₉₇	—	2	4	no	—
158	Dic₁₉₇	—	2	4	no	—
159	Dic₁₉₇	C₂²×D₄	1	8	yes	—
160	C₂×C₁₉₇	—	2	4	no	—
161	C₂×C₁₉₇	—	4	2	no	—
162	C₂×C₁₉₇	#548	1	4	yes	complemented
163	C₂×C₁₉₇	C₂³×D₄	1	1	yes	characteristic
164	C₂×C₁₉₇	C₂²×#41	1	1	yes	characteristic
165	C₂×C₁₉₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
166	D₁₉₇	—	4	4	no	—
167	C₁₉₇	C₂×#548	1	1	yes	characteristic, complemented, Sylow
168	C₂×#548	—	197	1	no	maximal, Sylow
169	C₂²×#41	—	197	1	no	—
170	C₂³×Q₈	—	197	1	no	—
171	#548	—	197	16	no	—
172	C₂×#124	—	197	2	no	—
173	C₂×#124	—	197	1	no	—
174	C₂×C₂²⋊Q₈	—	197	1	no	—
175	C₂×C₂²⋊Q₈	—	197	2	no	—
176	C₂×#121	—	197	2	no	—
177	C₂×#121	—	197	1	no	—
178	C₂×C₄×Q₈	—	197	1	no	—
179	C₂×C₄×D₄	—	197	2	no	—
180	C₂×C₄×D₄	D₁₉₇	1	1	yes	characteristic, complemented
181	C₂×#41	—	394	8	no	—
182	C₂×#41	—	197	8	no	—
183	C₂²×Q₈	—	394	8	no	—
184	C₂²×Q₈	—	197	1	no	—
185	C₂²×Q₈	—	197	2	no	—
186	C₂²×Q₈	—	197	8	no	—
187	C₂²×Q₈	—	197	1	no	—
188	C₂²×D₄	—	197	2	no	—
189	C₂²×D₄	—	197	2	no	—
190	C₂²×D₄	—	197	1	no	—
191	C₂²×D₄	C₂×D₁₉₇	1	1	yes	characteristic
192	C₂³×C₄	—	197	2	no	—
193	C₂³×C₄	—	197	2	no	—
194	C₂³×C₄	C₂×D₁₉₇	1	1	yes	characteristic
195	C₂³×C₄	C₂×D₁₉₇	1	1	yes	characteristic
196	#124	—	197	16	no	—
197	#124	—	197	8	no	—
198	C₂²⋊Q₈	—	197	8	no	—
199	C₂²⋊Q₈	—	197	16	no	—
200	#121	—	197	16	no	—
201	#121	—	197	8	no	—
202	C₄×Q₈	—	197	8	no	—
203	C₄×D₄	—	197	16	no	—
204	C₄×D₄	C₂×D₁₉₇	1	8	yes	complemented
205	C₂×C₄⋊C₄	—	197	2	no	—
206	C₂×C₄⋊C₄	—	197	2	no	—
207	C₂×C₄⋊C₄	—	197	1	no	—
208	C₂×C₄⋊C₄	C₂×D₁₉₇	1	1	yes	characteristic
209	C₂×C₂²⋊C₄	—	197	2	no	—
210	C₂×C₂²⋊C₄	—	197	2	no	—
211	C₂×C₂²⋊C₄	—	197	2	no	—
212	C₂×C₂²⋊C₄	—	197	2	no	—
213	C₂×C₂²⋊C₄	C₂×D₁₉₇	1	1	yes	characteristic
214	C₂×C₂²⋊C₄	C₂×D₁₉₇	1	1	yes	characteristic
215	C₂×C₄²	—	197	2	no	—
216	C₂×C₄²	C₂×D₁₉₇	1	1	yes	characteristic
217	C₂⁴	—	197	2	no	—
218	C₂⁴	C₂²×D₁₉₇	1	1	yes	characteristic
219	C₂⁴	C₂²×D₁₉₇	1	1	yes	characteristic
220	#41	—	394	32	no	—
221	C₂×Q₈	—	394	32	no	—
222	C₂×Q₈	—	394	8	no	—
223	C₂×Q₈	—	394	4	no	—
224	C₂×Q₈	—	197	8	no	—
225	C₂×Q₈	—	197	4	no	—
226	C₂×Q₈	—	197	8	no	—
227	C₂×Q₈	—	197	4	no	—
228	C₂×D₄	—	2	4	no	—
229	C₂×D₄	—	394	8	no	—
230	C₂×D₄	—	394	8	no	—
231	C₂×D₄	—	394	4	no	—
232	C₂×D₄	—	197	8	no	—
233	C₂×D₄	—	197	8	no	—
234	C₂×D₄	—	197	4	no	—
235	C₂×D₄	C₂²×D₁₉₇	1	4	yes	—
236	C₂²×C₄	—	2	4	no	—
237	C₂²×C₄	—	2	4	no	—
238	C₂²×C₄	—	394	8	no	—
239	C₂²×C₄	—	394	8	no	—
240	C₂²×C₄	—	197	8	no	—
241	C₂²×C₄	—	197	2	no	—
242	C₂²×C₄	—	197	2	no	—
243	C₂²×C₄	—	197	8	no	—
244	C₂²×C₄	—	197	2	no	—
245	C₂²×C₄	C₂²×D₁₉₇	1	4	yes	—
246	C₂²×C₄	C₂²×D₁₉₇	1	4	yes	—
247	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
248	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
249	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
250	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
251	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
252	C₄⋊C₄	—	197	8	no	—
253	C₄⋊C₄	—	197	8	no	—
254	C₄⋊C₄	—	197	4	no	—
255	C₄⋊C₄	C₂²×D₁₉₇	1	4	yes	—
256	C₂²⋊C₄	—	197	8	no	—
257	C₂²⋊C₄	—	197	8	no	—
258	C₂²⋊C₄	—	197	8	no	—
259	C₂²⋊C₄	—	197	8	no	—
260	C₂²⋊C₄	C₂²×D₁₉₇	1	4	yes	—
261	C₂²⋊C₄	C₂²×D₁₉₇	1	4	yes	—
262	C₄²	—	197	8	no	—
263	C₄²	C₂²×D₁₉₇	1	4	yes	—
264	C₂³	—	2	2	no	—
265	C₂³	—	2	2	no	—
266	C₂³	—	2	2	no	—
267	C₂³	—	2	2	no	—
268	C₂³	—	2	2	no	—
269	C₂³	—	394	4	no	—
270	C₂³	—	394	4	no	—
271	C₂³	—	394	4	no	—
272	C₂³	—	197	4	no	—
273	C₂³	#15691358	1	4	yes	complemented
274	C₂³	C₂³×D₁₉₇	1	2	yes	—
275	C₂³	C₂³×D₁₉₇	1	2	yes	—
276	C₂³	C₂³×D₁₉₇	1	1	yes	characteristic
277	Q₈	—	394	16	no	—
278	Q₈	—	394	8	no	—
279	Q₈	—	197	16	no	—
280	D₄	—	2	8	no	—
281	D₄	—	394	16	no	—
282	D₄	—	394	16	no	—
283	D₄	—	394	8	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	8	no	—
289	C₂×C₄	—	2	8	no	—
290	C₂×C₄	—	394	16	no	—
291	C₂×C₄	—	394	4	no	—
292	C₂×C₄	—	394	4	no	—
293	C₂×C₄	—	394	16	no	—
294	C₂×C₄	—	197	4	no	—
295	C₂×C₄	—	197	4	no	—
296	C₂×C₄	—	197	8	no	—
297	C₂×C₄	—	197	4	no	—
298	C₂×C₄	D₄×D₁₉₇	1	4	yes	—
299	C₂×C₄	C₂³×D₁₉₇	1	2	yes	—
300	C₂×C₄	C₂³×D₁₉₇	1	2	yes	—
301	C₂×C₄	C₂³×D₁₉₇	1	2	yes	—
302	C₂×C₄	C₂³×D₁₉₇	1	2	yes	—
303	C₂×C₄	C₂³×D₁₉₇	1	2	yes	—
304	C₂²	—	2	2	no	—
305	C₂²	—	2	2	no	—
306	C₂²	—	2	8	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	2	2	no	—
309	C₂²	—	2	2	no	—
310	C₂²	—	4	4	no	—
311	C₂²	—	788	8	no	—
312	C₂²	—	394	4	no	—
313	C₂²	—	394	4	no	—
314	C₂²	—	394	4	no	—
315	C₂²	C₂×D₄×D₁₉₇	1	2	yes	—
316	C₂²	C₂×#15691358	1	2	yes	—
317	C₂²	C₂×#15691358	1	4	yes	complemented
318	C₂²	#20263373	1	2	yes	—
319	C₂²	C₂⁴×D₁₉₇	1	1	yes	characteristic
320	C₄	—	2	2	no	—
321	C₄	—	2	2	no	—
322	C₄	—	2	2	no	—
323	C₄	—	2	2	no	—
324	C₄	—	394	4	no	—
325	C₄	—	394	4	no	—
326	C₄	—	197	8	no	—
327	C₄	C₂×D₄×D₁₉₇	1	4	yes	—
328	C₂	—	2	4	no	—
329	C₂	—	4	2	no	—
330	C₂	—	788	4	no	—
331	C₂	#27161738	1	4	yes	direct product factor
332	C₂	C₂²×D₄×D₁₉₇	1	1	yes	characteristic
333	C₂	C₂²×#15691358	1	1	yes	characteristic
334	C₂	C₂×#20263373	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₉₇
commutator, cosocle	C₂²×C₁₉₇	C₂⁵
Frattini	C₂²	C₂⁴×D₁₉₇
Fitting	C₂×C₄×D₄×C₁₉₇	C₂
socle	C₂³×C₁₉₇	C₂⁴
hypercenter	C₂×C₄×D₄	D₁₉₇
hypocenter	C₁₉₇	C₂×#548
LCS	G ▷ C₂²×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#15691358	C₂	semidirect
2. C₂³×Dic₃₉₄	C₂	semidirect
3. #27161738	C₂	direct (N × H)
4. C₂×#20263314	C₂	semidirect
5. C₂×#20263312	C₂	semidirect
6. C₂×#20263305	C₂	semidirect
7. C₂×#20263304	C₂	semidirect
8. C₂×#20263300	C₂	semidirect
9. C₂×#20263298	C₂	semidirect
10. C₂×C₄×Dic₃₉₄	C₂	semidirect
11. C₂×#20263286	C₂	semidirect
12. C₂×C₄×D₄×C₁₉₇	C₂	semidirect
14. C₂×#15691358	C₂²	semidirect
19. C₂²×Dic₃₉₄	C₂²	semidirect
25. #20263314	C₂²	semidirect
26. #20263312	C₂²	semidirect
27. #20263305	C₂²	semidirect
28. #20263304	C₂²	semidirect
29. #20263300	C₂²	semidirect
30. #20263298	C₂²	semidirect
31. C₄×Dic₃₉₄	C₂²	semidirect
32. #20263286	C₂²	semidirect
44. C₄×D₄×C₁₉₇	C₂²	semidirect
54. C₂×Dic₃₉₄	D₄	semidirect
102. Dic₃₉₄	C₂×D₄	semidirect
119. C₂³×C₁₉₇	#41	semidirect
145. C₂²×C₁₉₇	C₂×#41	semidirect
162. C₂×C₁₉₇	#548	semidirect
167. C₁₉₇	C₂×#548	semidirect
180. C₂×C₄×D₄	D₁₉₇	semidirect
204. C₄×D₄	C₂×D₁₉₇	semidirect
273. C₂³	#15691358	semidirect
317. C₂²	C₂×#15691358	semidirect
331. C₂	#27161738	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#15691358	2
C₂³×Dic₃₉₄	2
#27161738	2
C₂×#20263314	2
C₂×#20263312	2
C₂×#20263305	2
C₂×#20263304	2
C₂×#20263300	2
C₂×#20263298	2
C₂×C₄×Dic₃₉₄	2
C₂×#20263286	2
C₂×C₄×D₄×C₁₉₇	2
C₂×#548	197

Subgroups — C2×#27161738 #38756178

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#27161738 #38756178