Subgroups — C₂×C₆₇×#2096 #30702979

Order 17152.

1238 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#106×C₆₇	C₂	1	2	yes	maximal
2	C₂²×#103×C₆₇	C₂	1	2	yes	maximal
3	C₂×#391×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#384×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#2096	C₂	1	8	yes	maximal, direct product factor
6	C₂×#336×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
8	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
9	C₂²×C₄⋊C₄×C₆₇	—	2	2	no	—
10	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
11	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
12	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×#106×C₆₇	—	2	8	no	—
14	C₂×#106×C₆₇	C₂²	1	8	yes	—
15	C₂×#103×C₆₇	—	2	8	no	—
16	C₂×#103×C₆₇	C₂²	1	8	yes	—
17	#391×C₆₇	—	2	2	no	—
18	#391×C₆₇	C₂²	1	4	yes	—
19	#384×C₆₇	—	2	2	no	—
20	#384×C₆₇	C₂²	1	4	yes	—
21	C₂×#95×C₆₇	—	2	2	no	—
22	#336×C₆₇	C₂²	1	4	yes	—
23	C₂²×Q₈×C₆₇	—	2	1	no	—
24	C₂²×Q₈×C₆₇	—	2	2	no	—
25	C₂²×Q₈×C₆₇	—	2	4	no	—
26	C₂²×Q₈×C₆₇	—	2	4	no	—
27	C₂²×Q₈×C₆₇	Q₈	1	4	yes	—
28	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
29	C₂³×C₄×C₆₇	—	2	1	no	—
30	C₂³×C₄×C₆₇	—	2	2	no	—
31	C₂³×C₄×C₆₇	—	2	1	no	—
32	C₂³×C₄×C₆₇	D₄	1	2	yes	—
33	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₆₇	—	2	4	no	—
35	C₂²×C₈×C₆₇	—	2	4	no	—
36	C₂²×C₈×C₆₇	D₄	1	8	yes	—
37	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
38	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
39	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
40	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
41	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
42	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
43	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
44	C₂×C₄⋊C₄×C₆₇	—	4	8	no	—
45	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
46	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
47	C₂×C₄²×C₆₇	—	2	4	no	—
48	C₂×C₄²×C₆₇	D₄	1	2	yes	—
49	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
50	#106×C₆₇	—	2	16	no	—
51	#103×C₆₇	—	2	16	no	—
52	#95×C₆₇	—	2	8	no	—
53	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
54	C₂×Q₈×C₆₇	—	2	4	no	—
55	C₂×Q₈×C₆₇	—	2	2	no	—
56	C₂×Q₈×C₆₇	—	2	16	no	—
57	C₂×Q₈×C₆₇	—	2	4	no	—
58	C₂×Q₈×C₆₇	—	2	4	no	—
59	C₂×Q₈×C₆₇	—	4	2	no	—
60	C₂×Q₈×C₆₇	—	4	8	no	—
61	C₂×Q₈×C₆₇	—	4	2	no	—
62	C₂×Q₈×C₆₇	C₂×Q₈	1	4	yes	—
63	C₂²×C₄×C₆₇	—	2	4	no	—
64	C₂²×C₄×C₆₇	—	2	4	no	—
65	C₂²×C₄×C₆₇	—	2	4	no	—
66	C₂²×C₄×C₆₇	—	2	2	no	—
67	C₂²×C₄×C₆₇	—	2	4	no	—
68	C₂²×C₄×C₆₇	—	2	2	no	—
69	C₂²×C₄×C₆₇	—	4	2	no	—
70	C₂²×C₄×C₆₇	—	4	2	no	—
71	C₂²×C₄×C₆₇	—	4	4	no	—
72	C₂²×C₄×C₆₇	—	4	2	no	—
73	C₂²×C₄×C₆₇	—	4	8	no	—
74	C₂²×C₄×C₆₇	—	4	4	no	—
75	C₂²×C₄×C₆₇	SD₁₆	1	8	yes	—
76	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
77	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
79	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
80	C₂²×C₄×C₆₇	#41	1	4	yes	—
81	C₂²×C₄×C₆₇	#41	1	2	yes	—
82	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
83	C₂×C₈×C₆₇	—	2	16	no	—
84	C₂×C₈×C₆₇	—	2	4	no	—
85	C₂×C₈×C₆₇	—	2	4	no	—
86	C₂×C₈×C₆₇	C₂×D₄	1	8	yes	—
87	C₄⋊C₄×C₆₇	—	2	4	no	—
88	C₄⋊C₄×C₆₇	—	2	4	no	—
89	C₄⋊C₄×C₆₇	—	2	4	no	—
90	C₄⋊C₄×C₆₇	—	4	4	no	—
91	C₄⋊C₄×C₆₇	—	4	2	no	—
92	C₄⋊C₄×C₆₇	—	4	16	no	—
93	C₄²×C₆₇	—	2	4	no	—
94	C₄²×C₆₇	—	2	4	no	—
95	C₂³×C₆₇	C₂²⋊Q₈	1	2	yes	—
96	C₂³×C₆₇	#123	1	2	yes	—
97	C₂³×C₆₇	#127	1	2	yes	—
98	C₂³×C₆₇	C₂×SD₁₆	1	4	yes	—
99	C₂³×C₆₇	#137	1	4	yes	—
100	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
101	Q₈×C₆₇	—	2	8	no	—
102	Q₈×C₆₇	—	4	4	no	—
103	Q₈×C₆₇	—	4	8	no	—
104	C₂×C₄×C₆₇	—	2	4	no	—
105	C₂×C₄×C₆₇	—	2	4	no	—
106	C₂×C₄×C₆₇	—	2	16	no	—
107	C₂×C₄×C₆₇	—	2	4	no	—
108	C₂×C₄×C₆₇	—	4	2	no	—
109	C₂×C₄×C₆₇	—	4	2	no	—
110	C₂×C₄×C₆₇	—	4	8	no	—
111	C₂×C₄×C₆₇	—	4	4	no	—
112	C₂×C₄×C₆₇	—	4	2	no	—
113	C₂×C₄×C₆₇	—	4	8	no	—
114	C₂×C₄×C₆₇	—	4	4	no	—
115	C₂×C₄×C₆₇	—	8	8	no	—
116	C₂×C₄×C₆₇	—	8	4	no	—
117	C₂×C₄×C₆₇	C₂²⋊Q₈	1	8	yes	—
118	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
119	C₂×C₄×C₆₇	#123	1	8	yes	—
120	C₂×C₄×C₆₇	#123	1	2	yes	—
121	C₂×C₄×C₆₇	#127	1	2	yes	—
122	C₂×C₄×C₆₇	C₂×SD₁₆	1	8	yes	—
123	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
124	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
125	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
126	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
127	C₈×C₆₇	—	2	8	no	—
128	C₂²×C₆₇	#399	1	4	yes	—
129	C₂²×C₆₇	#475	1	8	yes	—
130	C₂²×C₆₇	#483	1	8	yes	—
131	C₂²×C₆₇	#492	1	4	yes	—
132	C₂²×C₆₇	#497	1	4	yes	—
133	C₂²×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
134	C₂²×C₆₇	C₂×#123	1	1	yes	characteristic
135	C₂²×C₆₇	C₂×#127	1	1	yes	characteristic
136	C₂²×C₆₇	C₂²×SD₁₆	1	2	yes	—
137	C₂²×C₆₇	C₂×#137	1	2	yes	—
138	C₄×C₆₇	—	2	8	no	—
139	C₄×C₆₇	—	4	4	no	—
140	C₄×C₆₇	—	8	4	no	—
141	C₄×C₆₇	—	8	2	no	—
142	C₄×C₆₇	#399	1	4	yes	—
143	C₄×C₆₇	C₂×C₂²⋊Q₈	1	2	yes	—
144	C₄×C₆₇	C₂×#123	1	2	yes	—
145	C₂×#2096	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
146	C₂×C₆₇	#2096	1	8	yes	direct product factor
147	C₂×C₆₇	C₂×#399	1	1	yes	characteristic
148	C₂×C₆₇	C₂×#475	1	2	yes	—
149	C₂×C₆₇	C₂×#483	1	2	yes	—
150	C₂×C₆₇	C₂×#492	1	1	yes	characteristic
151	C₂×C₆₇	C₂×#497	1	1	yes	characteristic
152	C₂²×#106	C₂×C₆₇	1	2	yes	—
153	C₂²×#103	C₂×C₆₇	1	2	yes	—
154	C₂×#391	C₂×C₆₇	1	1	yes	characteristic
155	C₂×#384	C₂×C₆₇	1	1	yes	characteristic
156	#2096	C₂×C₆₇	1	8	yes	direct product factor
157	C₂×#336	C₂×C₆₇	1	1	yes	characteristic
158	C₆₇	C₂×#2096	1	1	yes	characteristic, direct product factor, Sylow
159	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
160	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
161	C₂²×C₄⋊C₄	—	2	2	no	—
162	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
163	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
164	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
165	C₂×#106	—	2	8	no	—
166	C₂×#106	C₂²×C₆₇	1	8	yes	—
167	C₂×#103	—	2	8	no	—
168	C₂×#103	C₂²×C₆₇	1	8	yes	—
169	#391	—	2	2	no	—
170	#391	C₂²×C₆₇	1	4	yes	—
171	#384	—	2	2	no	—
172	#384	C₂²×C₆₇	1	4	yes	—
173	C₂×#95	—	2	2	no	—
174	#336	C₂²×C₆₇	1	4	yes	—
175	C₂²×Q₈	—	2	1	no	—
176	C₂²×Q₈	—	2	2	no	—
177	C₂²×Q₈	—	2	4	no	—
178	C₂²×Q₈	—	2	4	no	—
179	C₂²×Q₈	Q₈×C₆₇	1	4	yes	—
180	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
181	C₂³×C₄	—	2	1	no	—
182	C₂³×C₄	—	2	2	no	—
183	C₂³×C₄	—	2	1	no	—
184	C₂³×C₄	D₄×C₆₇	1	2	yes	—
185	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
186	C₂²×C₈	—	2	4	no	—
187	C₂²×C₈	—	2	4	no	—
188	C₂²×C₈	D₄×C₆₇	1	8	yes	—
189	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
190	C₂×C₄⋊C₄	—	2	4	no	—
191	C₂×C₄⋊C₄	—	2	4	no	—
192	C₂×C₄⋊C₄	—	2	4	no	—
193	C₂×C₄⋊C₄	—	2	8	no	—
194	C₂×C₄⋊C₄	—	2	2	no	—
195	C₂×C₄⋊C₄	—	2	1	no	—
196	C₂×C₄⋊C₄	—	4	8	no	—
197	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
198	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
199	C₂×C₄²	—	2	4	no	—
200	C₂×C₄²	D₄×C₆₇	1	2	yes	—
201	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
202	#106	—	2	16	no	—
203	#103	—	2	16	no	—
204	#95	—	2	8	no	—
205	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
206	C₂×Q₈	—	2	4	no	—
207	C₂×Q₈	—	2	2	no	—
208	C₂×Q₈	—	2	16	no	—
209	C₂×Q₈	—	2	4	no	—
210	C₂×Q₈	—	2	4	no	—
211	C₂×Q₈	—	4	2	no	—
212	C₂×Q₈	—	4	8	no	—
213	C₂×Q₈	—	4	2	no	—
214	C₂×Q₈	C₂×Q₈×C₆₇	1	4	yes	—
215	C₂²×C₄	—	2	4	no	—
216	C₂²×C₄	—	2	4	no	—
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	4	no	—
221	C₂²×C₄	—	4	2	no	—
222	C₂²×C₄	—	4	2	no	—
223	C₂²×C₄	—	4	4	no	—
224	C₂²×C₄	—	4	2	no	—
225	C₂²×C₄	—	4	8	no	—
226	C₂²×C₄	—	4	4	no	—
227	C₂²×C₄	SD₁₆×C₆₇	1	8	yes	—
228	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
229	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
230	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
231	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
232	C₂²×C₄	#41×C₆₇	1	4	yes	—
233	C₂²×C₄	#41×C₆₇	1	2	yes	—
234	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
235	C₂×C₈	—	2	16	no	—
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	C₂×D₄×C₆₇	1	8	yes	—
239	C₄⋊C₄	—	2	4	no	—
240	C₄⋊C₄	—	2	4	no	—
241	C₄⋊C₄	—	2	4	no	—
242	C₄⋊C₄	—	4	4	no	—
243	C₄⋊C₄	—	4	2	no	—
244	C₄⋊C₄	—	4	16	no	—
245	C₄²	—	2	4	no	—
246	C₄²	—	2	4	no	—
247	C₂³	C₂²⋊Q₈×C₆₇	1	2	yes	—
248	C₂³	#123×C₆₇	1	2	yes	—
249	C₂³	#127×C₆₇	1	2	yes	—
250	C₂³	C₂×SD₁₆×C₆₇	1	4	yes	—
251	C₂³	#137×C₆₇	1	4	yes	—
252	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
253	Q₈	—	2	8	no	—
254	Q₈	—	4	4	no	—
255	Q₈	—	4	8	no	—
256	C₂×C₄	—	2	16	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	8	no	—
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	4	8	no	—
266	C₂×C₄	—	4	4	no	—
267	C₂×C₄	—	8	8	no	—
268	C₂×C₄	—	8	4	no	—
269	C₂×C₄	C₂²⋊Q₈×C₆₇	1	8	yes	—
270	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
271	C₂×C₄	#123×C₆₇	1	8	yes	—
272	C₂×C₄	#123×C₆₇	1	2	yes	—
273	C₂×C₄	#127×C₆₇	1	2	yes	—
274	C₂×C₄	C₂×SD₁₆×C₆₇	1	8	yes	—
275	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
276	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
277	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
278	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
279	C₈	—	2	8	no	—
280	C₂²	#399×C₆₇	1	4	yes	—
281	C₂²	#475×C₆₇	1	8	yes	—
282	C₂²	#483×C₆₇	1	8	yes	—
283	C₂²	#492×C₆₇	1	4	yes	—
284	C₂²	#497×C₆₇	1	4	yes	—
285	C₂²	C₂×C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
286	C₂²	C₂×#123×C₆₇	1	1	yes	characteristic
287	C₂²	C₂×#127×C₆₇	1	1	yes	characteristic
288	C₂²	C₂²×SD₁₆×C₆₇	1	2	yes	—
289	C₂²	C₂×#137×C₆₇	1	2	yes	—
290	C₄	—	2	8	no	—
291	C₄	—	4	4	no	—
292	C₄	—	8	4	no	—
293	C₄	—	8	2	no	—
294	C₄	#399×C₆₇	1	4	yes	—
295	C₄	C₂×C₂²⋊Q₈×C₆₇	1	2	yes	—
296	C₄	C₂×#123×C₆₇	1	2	yes	—
297	C₂	C₆₇×#2096	1	8	yes	direct product factor
298	C₂	C₂×#399×C₆₇	1	1	yes	characteristic
299	C₂	C₂×#475×C₆₇	1	2	yes	—
300	C₂	C₂×#483×C₆₇	1	2	yes	—
301	C₂	C₂×#492×C₆₇	1	1	yes	characteristic
302	C₂	C₂×#497×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₆₇	C₂×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₆₇×#2096	C₂	direct (N × H)
145. C₂×#2096	C₆₇	direct (N × H)
146. C₂×C₆₇	#2096	direct (N × H)
156. #2096	C₂×C₆₇	direct (N × H)
158. C₆₇	C₂×#2096	direct (N × H)
297. C₂	C₆₇×#2096	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×#106×C₆₇	2
C₂²×#103×C₆₇	2
C₂×#391×C₆₇	2
C₂×#384×C₆₇	2
C₆₇×#2096	2
C₂×#336×C₆₇	2
C₂×#2096	67

Subgroups — C2×C67×#2096 #30702979

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2096 #30702979