Subgroups — C₂×C₁₀₇×#2234 #41059605

Order 27392.

638 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#466×C₁₀₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#427×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₀₇×#2234	C₂	1	8	yes	maximal, direct product factor
4	C₂×#367×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#366×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#357×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#357×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂²⋊C₁₆×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₈×C₁₀₇	C₂²	1	1	yes	characteristic
10	C₂³×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
11	C₂×#121×C₁₀₇	—	2	1	no	—
12	C₂×#117×C₁₀₇	—	2	1	no	—
13	C₂²×C₁₆×C₁₀₇	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆×C₁₀₇	C₂²	1	1	yes	characteristic
15	#466×C₁₀₇	—	2	2	no	—
16	#466×C₁₀₇	C₂²	1	4	yes	—
17	#427×C₁₀₇	—	2	2	no	—
18	#427×C₁₀₇	C₂²	1	4	yes	complemented
19	C₂×#107×C₁₀₇	C₂²	1	1	yes	characteristic
20	C₂×#107×C₁₀₇	C₂²	1	1	yes	characteristic
21	C₂×#107×C₁₀₇	C₂²	1	1	yes	characteristic
22	C₂×#106×C₁₀₇	—	2	1	no	—
23	C₂×#102×C₁₀₇	—	2	1	no	—
24	#367×C₁₀₇	C₂²	1	4	yes	complemented
25	#366×C₁₀₇	C₂²	1	4	yes	complemented
26	#357×C₁₀₇	C₂²	1	4	yes	complemented
27	#357×C₁₀₇	C₂²	1	4	yes	complemented
28	C₂²⋊C₁₆×C₁₀₇	C₂²	1	4	yes	complemented
29	C₂²×D₄×C₁₀₇	—	2	1	no	—
30	C₂²×D₄×C₁₀₇	—	4	1	no	—
31	C₂³×C₄×C₁₀₇	D₄	1	1	yes	characteristic
32	C₂×D₈×C₁₀₇	—	2	1	no	—
33	C₂×D₈×C₁₀₇	—	2	2	no	—
34	C₂×D₈×C₁₀₇	—	2	2	no	—
35	C₂×D₈×C₁₀₇	C₂³	1	2	yes	—
36	C₂²×C₈×C₁₀₇	—	2	2	no	—
37	C₂²×C₈×C₁₀₇	—	2	2	no	—
38	C₂²×C₈×C₁₀₇	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₁₀₇	D₄	1	2	yes	—
40	C₂²×C₈×C₁₀₇	C₂³	1	2	yes	—
41	C₂²×C₈×C₁₀₇	C₂³	1	1	yes	characteristic
42	#121×C₁₀₇	—	2	4	no	—
43	#121×C₁₀₇	—	4	2	no	—
44	#117×C₁₀₇	—	2	4	no	—
45	#117×C₁₀₇	—	4	2	no	—
46	C₂×C₄⋊C₄×C₁₀₇	—	2	1	no	—
47	C₂×C₄⋊C₄×C₁₀₇	—	2	1	no	—
48	C₂×C₄⋊C₄×C₁₀₇	—	2	1	no	—
49	C₂×C₂²⋊C₄×C₁₀₇	—	4	1	no	—
50	C₂×C₂²⋊C₄×C₁₀₇	—	4	1	no	—
51	C₂×C₄²×C₁₀₇	—	4	1	no	—
52	C₂×C₁₆×C₁₀₇	—	2	2	no	—
53	C₂×C₁₆×C₁₀₇	—	2	2	no	—
54	C₂×C₁₆×C₁₀₇	C₂³	1	2	yes	—
55	C₂×C₁₆×C₁₀₇	C₂³	1	2	yes	—
56	#107×C₁₀₇	—	2	1	no	—
57	#107×C₁₀₇	—	2	1	no	—
58	#107×C₁₀₇	—	2	1	no	—
59	#107×C₁₀₇	C₂³	1	2	yes	—
60	#107×C₁₀₇	C₂³	1	2	yes	—
61	#107×C₁₀₇	C₂³	1	2	yes	—
62	#106×C₁₀₇	—	2	4	no	—
63	#102×C₁₀₇	—	2	4	no	—
64	C₂⁴×C₁₀₇	—	4	1	no	—
65	C₂⁴×C₁₀₇	D₈	1	1	yes	characteristic
66	C₂×D₄×C₁₀₇	—	2	2	no	—
67	C₂×D₄×C₁₀₇	—	4	4	no	—
68	C₂×D₄×C₁₀₇	—	4	2	no	—
69	C₂×D₄×C₁₀₇	—	4	1	no	—
70	C₂×D₄×C₁₀₇	—	4	2	no	—
71	C₂×D₄×C₁₀₇	—	8	4	no	—
72	C₂²×C₄×C₁₀₇	—	2	2	no	—
73	C₂²×C₄×C₁₀₇	—	2	2	no	—
74	C₂²×C₄×C₁₀₇	—	4	1	no	—
75	C₂²×C₄×C₁₀₇	—	4	1	no	—
76	C₂²×C₄×C₁₀₇	—	4	1	no	—
77	C₂²×C₄×C₁₀₇	D₈	1	2	yes	—
78	C₂²×C₄×C₁₀₇	D₈	1	1	yes	characteristic
79	C₂²×C₄×C₁₀₇	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
81	D₈×C₁₀₇	—	2	2	no	—
82	D₈×C₁₀₇	—	2	2	no	—
83	D₈×C₁₀₇	—	4	2	no	—
84	C₂×C₈×C₁₀₇	—	2	2	no	—
85	C₂×C₈×C₁₀₇	—	2	4	no	—
86	C₂×C₈×C₁₀₇	—	2	2	no	—
87	C₂×C₈×C₁₀₇	—	2	2	no	—
88	C₂×C₈×C₁₀₇	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₁₀₇	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₁₀₇	#41	1	2	yes	—
91	C₂×C₈×C₁₀₇	#41	1	2	yes	—
92	C₂×C₈×C₁₀₇	C₂⁴	1	1	yes	characteristic
93	C₄⋊C₄×C₁₀₇	—	2	2	no	—
94	C₄⋊C₄×C₁₀₇	—	2	2	no	—
95	C₄⋊C₄×C₁₀₇	—	2	2	no	—
96	C₄⋊C₄×C₁₀₇	—	4	1	no	—
97	C₄⋊C₄×C₁₀₇	—	4	1	no	—
98	C₄⋊C₄×C₁₀₇	—	4	1	no	—
99	C₂²⋊C₄×C₁₀₇	—	4	4	no	—
100	C₂²⋊C₄×C₁₀₇	—	4	4	no	—
101	C₄²×C₁₀₇	—	4	4	no	—
102	C₁₆×C₁₀₇	—	2	2	no	—
103	C₁₆×C₁₀₇	—	2	2	no	—
104	C₂³×C₁₀₇	—	2	2	no	—
105	C₂³×C₁₀₇	—	2	2	no	—
106	C₂³×C₁₀₇	—	2	2	no	—
107	C₂³×C₁₀₇	—	4	2	no	—
108	C₂³×C₁₀₇	—	8	2	no	—
109	C₂³×C₁₀₇	—	8	2	no	—
110	C₂³×C₁₀₇	—	8	2	no	—
111	C₂³×C₁₀₇	C₂×D₈	1	2	yes	—
112	C₂³×C₁₀₇	C₂×D₈	1	1	yes	characteristic
113	D₄×C₁₀₇	—	4	2	no	—
114	D₄×C₁₀₇	—	4	2	no	—
115	D₄×C₁₀₇	—	8	8	no	—
116	D₄×C₁₀₇	—	8	2	no	—
117	C₂×C₄×C₁₀₇	—	2	2	no	—
118	C₂×C₄×C₁₀₇	—	2	2	no	—
119	C₂×C₄×C₁₀₇	—	2	2	no	—
120	C₂×C₄×C₁₀₇	—	2	4	no	—
121	C₂×C₄×C₁₀₇	—	4	2	no	—
122	C₂×C₄×C₁₀₇	—	4	2	no	—
123	C₂×C₄×C₁₀₇	—	4	2	no	—
124	C₂×C₄×C₁₀₇	—	8	2	no	—
125	C₂×C₄×C₁₀₇	—	8	2	no	—
126	C₂×C₄×C₁₀₇	—	8	2	no	—
127	C₂×C₄×C₁₀₇	#123	1	2	yes	—
128	C₂×C₄×C₁₀₇	C₂×D₈	1	2	yes	—
129	C₂×C₄×C₁₀₇	C₂×D₈	1	1	yes	characteristic
130	C₂×C₄×C₁₀₇	#137	1	2	yes	—
131	C₂×C₄×C₁₀₇	C₂²×D₄	1	1	yes	characteristic
132	C₈×C₁₀₇	—	2	2	no	—
133	C₈×C₁₀₇	#123	1	2	yes	—
134	C₈×C₁₀₇	C₂×#41	1	1	yes	characteristic
135	C₈×C₁₀₇	C₂×#41	1	1	yes	characteristic
136	C₂²×C₁₀₇	—	2	2	no	—
137	C₂²×C₁₀₇	—	2	2	no	—
138	C₂²×C₁₀₇	—	2	2	no	—
139	C₂²×C₁₀₇	—	4	4	no	—
140	C₂²×C₁₀₇	—	8	2	no	—
141	C₂²×C₁₀₇	—	8	2	no	—
142	C₂²×C₁₀₇	—	8	2	no	—
143	C₂²×C₁₀₇	—	16	4	no	—
144	C₂²×C₁₀₇	#480	1	2	yes	—
145	C₂²×C₁₀₇	#508	1	2	yes	—
146	C₂²×C₁₀₇	#509	1	2	yes	—
147	C₂²×C₁₀₇	C₂²×D₈	1	1	yes	characteristic
148	C₄×C₁₀₇	—	2	2	no	—
149	C₄×C₁₀₇	—	8	2	no	—
150	C₄×C₁₀₇	—	8	2	no	—
151	C₄×C₁₀₇	—	8	2	no	—
152	C₄×C₁₀₇	#480	1	2	yes	—
153	C₄×C₁₀₇	C₂×#123	1	1	yes	characteristic
154	C₄×C₁₀₇	C₂×#137	1	1	yes	characteristic
155	C₂×#2234	C₁₀₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
156	C₂×C₁₀₇	—	4	2	no	—
157	C₂×C₁₀₇	—	16	2	no	—
158	C₂×C₁₀₇	#2234	1	4	yes	direct product factor
159	C₂×C₁₀₇	C₂×#480	1	1	yes	characteristic
160	C₂×C₁₀₇	C₂×#508	1	1	yes	characteristic
161	C₂×C₁₀₇	C₂×#509	1	1	yes	characteristic
162	C₂×#466	C₂×C₁₀₇	1	1	yes	characteristic
163	C₂×#427	C₂×C₁₀₇	1	1	yes	characteristic, complemented
164	#2234	C₂×C₁₀₇	1	8	yes	direct product factor
165	C₂×#367	C₂×C₁₀₇	1	1	yes	characteristic, complemented
166	C₂×#366	C₂×C₁₀₇	1	1	yes	characteristic, complemented
167	C₂×#357	C₂×C₁₀₇	1	1	yes	characteristic, complemented
168	C₂×#357	C₂×C₁₀₇	1	1	yes	characteristic, complemented
169	C₂×C₂²⋊C₁₆	C₂×C₁₀₇	1	1	yes	characteristic, complemented
170	C₁₀₇	C₂×#2234	1	1	yes	characteristic, direct product factor, Sylow
171	C₂²×D₈	C₂²×C₁₀₇	1	1	yes	characteristic
172	C₂³×C₈	C₂²×C₁₀₇	1	1	yes	characteristic
173	C₂×#121	—	2	1	no	—
174	C₂×#117	—	2	1	no	—
175	C₂²×C₁₆	C₂²×C₁₀₇	1	1	yes	characteristic
176	C₂²×C₁₆	C₂²×C₁₀₇	1	1	yes	characteristic
177	#466	—	2	2	no	—
178	#466	C₂²×C₁₀₇	1	4	yes	—
179	#427	—	2	2	no	—
180	#427	C₂²×C₁₀₇	1	4	yes	complemented
181	C₂×#107	C₂²×C₁₀₇	1	1	yes	characteristic
182	C₂×#107	C₂²×C₁₀₇	1	1	yes	characteristic
183	C₂×#107	C₂²×C₁₀₇	1	1	yes	characteristic
184	C₂×#106	—	2	1	no	—
185	C₂×#102	—	2	1	no	—
186	#367	C₂²×C₁₀₇	1	4	yes	complemented
187	#366	C₂²×C₁₀₇	1	4	yes	complemented
188	#357	C₂²×C₁₀₇	1	4	yes	complemented
189	#357	C₂²×C₁₀₇	1	4	yes	complemented
190	C₂²⋊C₁₆	C₂²×C₁₀₇	1	4	yes	complemented
191	C₂²×D₄	—	2	1	no	—
192	C₂²×D₄	—	4	1	no	—
193	C₂³×C₄	D₄×C₁₀₇	1	1	yes	characteristic
194	C₂×D₈	—	2	1	no	—
195	C₂×D₈	—	2	2	no	—
196	C₂×D₈	—	2	2	no	—
197	C₂×D₈	C₂³×C₁₀₇	1	2	yes	—
198	C₂²×C₈	—	2	2	no	—
199	C₂²×C₈	—	2	2	no	—
200	C₂²×C₈	D₄×C₁₀₇	1	1	yes	characteristic
201	C₂²×C₈	D₄×C₁₀₇	1	2	yes	—
202	C₂²×C₈	C₂³×C₁₀₇	1	2	yes	—
203	C₂²×C₈	C₂³×C₁₀₇	1	1	yes	characteristic
204	#121	—	2	4	no	—
205	#121	—	4	2	no	—
206	#117	—	2	4	no	—
207	#117	—	4	2	no	—
208	C₂×C₄⋊C₄	—	2	1	no	—
209	C₂×C₄⋊C₄	—	2	1	no	—
210	C₂×C₄⋊C₄	—	2	1	no	—
211	C₂×C₂²⋊C₄	—	4	1	no	—
212	C₂×C₂²⋊C₄	—	4	1	no	—
213	C₂×C₄²	—	4	1	no	—
214	C₂×C₁₆	—	2	2	no	—
215	C₂×C₁₆	—	2	2	no	—
216	C₂×C₁₆	C₂³×C₁₀₇	1	2	yes	—
217	C₂×C₁₆	C₂³×C₁₀₇	1	2	yes	—
218	#107	—	2	1	no	—
219	#107	—	2	1	no	—
220	#107	—	2	1	no	—
221	#107	C₂³×C₁₀₇	1	2	yes	—
222	#107	C₂³×C₁₀₇	1	2	yes	—
223	#107	C₂³×C₁₀₇	1	2	yes	—
224	#106	—	2	4	no	—
225	#102	—	2	4	no	—
226	C₂⁴	—	4	1	no	—
227	C₂⁴	D₈×C₁₀₇	1	1	yes	characteristic
228	C₂×D₄	—	2	2	no	—
229	C₂×D₄	—	4	4	no	—
230	C₂×D₄	—	4	2	no	—
231	C₂×D₄	—	4	1	no	—
232	C₂×D₄	—	4	2	no	—
233	C₂×D₄	—	8	4	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	4	1	no	—
237	C₂²×C₄	—	4	1	no	—
238	C₂²×C₄	—	4	1	no	—
239	C₂²×C₄	D₈×C₁₀₇	1	1	yes	characteristic
240	C₂²×C₄	D₈×C₁₀₇	1	2	yes	—
241	C₂²×C₄	C₂×D₄×C₁₀₇	1	2	yes	—
242	C₂²×C₄	C₂×D₄×C₁₀₇	1	1	yes	characteristic
243	D₈	—	2	2	no	—
244	D₈	—	2	2	no	—
245	D₈	—	4	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	4	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	C₂×D₄×C₁₀₇	1	2	yes	—
251	C₂×C₈	C₂×D₄×C₁₀₇	1	1	yes	characteristic
252	C₂×C₈	#41×C₁₀₇	1	2	yes	—
253	C₂×C₈	#41×C₁₀₇	1	2	yes	—
254	C₂×C₈	C₂⁴×C₁₀₇	1	1	yes	characteristic
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	4	1	no	—
259	C₄⋊C₄	—	4	1	no	—
260	C₄⋊C₄	—	4	1	no	—
261	C₂²⋊C₄	—	4	4	no	—
262	C₂²⋊C₄	—	4	4	no	—
263	C₄²	—	4	4	no	—
264	C₁₆	—	2	2	no	—
265	C₁₆	—	2	2	no	—
266	C₂³	—	2	2	no	—
267	C₂³	—	2	2	no	—
268	C₂³	—	2	2	no	—
269	C₂³	—	4	2	no	—
270	C₂³	—	8	2	no	—
271	C₂³	—	8	2	no	—
272	C₂³	—	8	2	no	—
273	C₂³	C₂×D₈×C₁₀₇	1	2	yes	—
274	C₂³	C₂×D₈×C₁₀₇	1	1	yes	characteristic
275	D₄	—	4	2	no	—
276	D₄	—	4	2	no	—
277	D₄	—	8	8	no	—
278	D₄	—	8	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	8	2	no	—
287	C₂×C₄	—	8	2	no	—
288	C₂×C₄	—	8	2	no	—
289	C₂×C₄	#123×C₁₀₇	1	2	yes	—
290	C₂×C₄	C₂×D₈×C₁₀₇	1	2	yes	—
291	C₂×C₄	C₂×D₈×C₁₀₇	1	1	yes	characteristic
292	C₂×C₄	#137×C₁₀₇	1	2	yes	—
293	C₂×C₄	C₂²×D₄×C₁₀₇	1	1	yes	characteristic
294	C₈	—	2	2	no	—
295	C₈	#123×C₁₀₇	1	2	yes	—
296	C₈	C₂×#41×C₁₀₇	1	1	yes	characteristic
297	C₈	C₂×#41×C₁₀₇	1	1	yes	characteristic
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	4	4	no	—
302	C₂²	—	8	2	no	—
303	C₂²	—	8	2	no	—
304	C₂²	—	8	2	no	—
305	C₂²	—	16	4	no	—
306	C₂²	#480×C₁₀₇	1	2	yes	—
307	C₂²	#508×C₁₀₇	1	2	yes	—
308	C₂²	#509×C₁₀₇	1	2	yes	—
309	C₂²	C₂²×D₈×C₁₀₇	1	1	yes	characteristic
310	C₄	—	2	2	no	—
311	C₄	—	8	2	no	—
312	C₄	—	8	2	no	—
313	C₄	—	8	2	no	—
314	C₄	#480×C₁₀₇	1	2	yes	—
315	C₄	C₂×#123×C₁₀₇	1	1	yes	characteristic
316	C₄	C₂×#137×C₁₀₇	1	1	yes	characteristic
317	C₂	—	4	2	no	—
318	C₂	—	16	2	no	—
319	C₂	C₁₀₇×#2234	1	4	yes	direct product factor
320	C₂	C₂×#480×C₁₀₇	1	1	yes	characteristic
321	C₂	C₂×#508×C₁₀₇	1	1	yes	characteristic
322	C₂	C₂×#509×C₁₀₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₀₇	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₀₇
prime radical	C₂×#466×C₁₀₇	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₇ ◁ C₂³×C₄×C₁₀₇ ◁ C₂³×C₈×C₁₀₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#427×C₁₀₇	C₂	semidirect
3. C₁₀₇×#2234	C₂	direct (N × H)
4. C₂×#367×C₁₀₇	C₂	semidirect
5. C₂×#366×C₁₀₇	C₂	semidirect
6. C₂×#357×C₁₀₇	C₂	semidirect
7. C₂×#357×C₁₀₇	C₂	semidirect
8. C₂×C₂²⋊C₁₆×C₁₀₇	C₂	semidirect
18. #427×C₁₀₇	C₂²	semidirect
24. #367×C₁₀₇	C₂²	semidirect
25. #366×C₁₀₇	C₂²	semidirect
26. #357×C₁₀₇	C₂²	semidirect
27. #357×C₁₀₇	C₂²	semidirect
28. C₂²⋊C₁₆×C₁₀₇	C₂²	semidirect
155. C₂×#2234	C₁₀₇	direct (N × H)
158. C₂×C₁₀₇	#2234	direct (N × H)
163. C₂×#427	C₂×C₁₀₇	semidirect
164. #2234	C₂×C₁₀₇	direct (N × H)
165. C₂×#367	C₂×C₁₀₇	semidirect
166. C₂×#366	C₂×C₁₀₇	semidirect
167. C₂×#357	C₂×C₁₀₇	semidirect
168. C₂×#357	C₂×C₁₀₇	semidirect
169. C₂×C₂²⋊C₁₆	C₂×C₁₀₇	semidirect
170. C₁₀₇	C₂×#2234	direct (N × H)
180. #427	C₂²×C₁₀₇	semidirect
186. #367	C₂²×C₁₀₇	semidirect
187. #366	C₂²×C₁₀₇	semidirect
188. #357	C₂²×C₁₀₇	semidirect
189. #357	C₂²×C₁₀₇	semidirect
190. C₂²⋊C₁₆	C₂²×C₁₀₇	semidirect
319. C₂	C₁₀₇×#2234	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#466×C₁₀₇	2
C₂×#427×C₁₀₇	2
C₁₀₇×#2234	2
C₂×#367×C₁₀₇	2
C₂×#366×C₁₀₇	2
C₂×#357×C₁₀₇	2
C₂×C₂²⋊C₁₆×C₁₀₇	2
C₂×#2234	107

Subgroups — C2×C107×#2234 #41059605

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₀₇×#2234 #41059605