Subgroups — C₂×C₆₇×#2310 #30713293

Order 17152.

5214 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×#117×C₆₇	C₂	1	2	yes	maximal
3	C₆₇×#2310	C₂	1	16	yes	maximal, direct product factor
4	C₂×#391×C₆₇	C₂	1	8	yes	maximal, complemented
5	C₂×#381×C₆₇	C₂	1	4	yes	maximal
6	C₂³×Q₈×C₆₇	—	2	16	no	—
7	C₂³×Q₈×C₆₇	C₄	1	16	yes	complemented
8	C₂³×Q₈×C₆₇	C₂²	1	4	yes	—
9	C₂³×Q₈×C₆₇	C₂²	1	8	yes	—
10	C₂⁴×C₄×C₆₇	C₂²	1	2	yes	—
11	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×#117×C₆₇	—	2	16	no	—
13	C₂×#117×C₆₇	C₂²	1	16	yes	—
14	C₂²×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
15	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
16	C₂²×C₄²×C₆₇	C₂²	1	4	yes	—
17	#391×C₆₇	C₂²	1	64	yes	complemented
18	#381×C₆₇	C₂²	1	32	yes	—
19	C₂×#95×C₆₇	C₂²	1	16	yes	—
20	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
21	C₂²×Q₈×C₆₇	—	2	128	no	—
22	C₂²×Q₈×C₆₇	—	2	16	no	—
23	C₂²×Q₈×C₆₇	—	2	16	no	—
24	C₂²×Q₈×C₆₇	—	2	64	no	—
25	C₂²×Q₈×C₆₇	C₂×C₄	1	32	yes	—
26	C₂²×Q₈×C₆₇	C₂×C₄	1	64	yes	complemented
27	C₂²×Q₈×C₆₇	C₂³	1	16	yes	—
28	C₂³×C₄×C₆₇	—	2	16	no	—
29	C₂³×C₄×C₆₇	C₂×C₄	1	16	yes	—
30	C₂³×C₄×C₆₇	C₂×C₄	1	8	yes	—
31	C₂³×C₄×C₆₇	D₄	1	16	yes	—
32	C₂³×C₄×C₆₇	C₂³	1	8	yes	—
33	C₂³×C₄×C₆₇	C₂³	1	8	yes	—
34	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
35	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
36	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
37	#117×C₆₇	—	2	64	no	—
38	C₂×C₄⋊C₄×C₆₇	—	2	16	no	—
39	C₂×C₄⋊C₄×C₆₇	C₂³	1	16	yes	—
40	C₂×C₂²⋊C₄×C₆₇	—	2	16	no	—
41	C₂×C₂²⋊C₄×C₆₇	C₂³	1	16	yes	—
42	C₂×C₄²×C₆₇	—	2	16	no	—
43	C₂×C₄²×C₆₇	C₂³	1	16	yes	—
44	#95×C₆₇	C₂³	1	64	yes	—
45	C₂⁴×C₆₇	—	2	8	no	—
46	C₂⁴×C₆₇	C₂²×C₄	1	4	yes	—
47	C₂⁴×C₆₇	C₂×D₄	1	8	yes	—
48	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
49	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
50	C₂×Q₈×C₆₇	—	2	128	no	—
51	C₂×Q₈×C₆₇	—	2	256	no	—
52	C₂×Q₈×C₆₇	—	2	32	no	—
53	C₂×Q₈×C₆₇	—	2	32	no	—
54	C₂×Q₈×C₆₇	C₂²×C₄	1	64	yes	—
55	C₂²×C₄×C₆₇	—	2	64	no	—
56	C₂²×C₄×C₆₇	—	2	32	no	—
57	C₂²×C₄×C₆₇	—	2	16	no	—
58	C₂²×C₄×C₆₇	—	2	16	no	—
59	C₂²×C₄×C₆₇	—	2	32	no	—
60	C₂²×C₄×C₆₇	C₂²⋊C₄	1	64	yes	—
61	C₂²×C₄×C₆₇	C₂²×C₄	1	16	yes	—
62	C₂²×C₄×C₆₇	C₂²×C₄	1	32	yes	—
63	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
64	C₂²×C₄×C₆₇	C₂²×C₄	1	16	yes	—
65	C₂²×C₄×C₆₇	C₂×D₄	1	8	yes	—
66	C₂²×C₄×C₆₇	C₂×D₄	1	8	yes	—
67	C₂²×C₄×C₆₇	C₂×D₄	1	32	yes	—
68	C₂²×C₄×C₆₇	C₂⁴	1	16	yes	—
69	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
70	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
71	C₄⋊C₄×C₆₇	—	2	32	no	—
72	C₂²⋊C₄×C₆₇	—	2	32	no	—
73	C₄²×C₆₇	—	2	32	no	—
74	C₂³×C₆₇	—	2	16	no	—
75	C₂³×C₆₇	—	2	32	no	—
76	C₂³×C₆₇	—	2	8	no	—
77	C₂³×C₆₇	C₂×C₂²⋊C₄	1	16	yes	—
78	C₂³×C₆₇	C₂³×C₄	1	4	yes	—
79	C₂³×C₆₇	C₂³×C₄	1	2	yes	—
80	C₂³×C₆₇	C₂²×D₄	1	8	yes	—
81	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
82	C₂³×C₆₇	2₋¹⁺⁴	1	8	yes	—
83	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
84	Q₈×C₆₇	—	2	128	no	—
85	C₂×C₄×C₆₇	—	2	64	no	—
86	C₂×C₄×C₆₇	—	2	32	no	—
87	C₂×C₄×C₆₇	—	2	16	no	—
88	C₂×C₄×C₆₇	—	2	16	no	—
89	C₂×C₄×C₆₇	—	2	32	no	—
90	C₂×C₄×C₆₇	—	4	32	no	—
91	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	64	yes	—
92	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	32	yes	—
93	C₂×C₄×C₆₇	C₂³×C₄	1	16	yes	—
94	C₂×C₄×C₆₇	C₂³×C₄	1	8	yes	—
95	C₂×C₄×C₆₇	C₂²×D₄	1	8	yes	—
96	C₂×C₄×C₆₇	C₂²×D₄	1	8	yes	—
97	C₂²×C₆₇	—	2	32	no	—
98	C₂²×C₆₇	—	2	8	no	—
99	C₂²×C₆₇	—	2	16	no	—
100	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	8	yes	—
101	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	4	yes	—
102	C₂²×C₆₇	#519	1	8	yes	—
103	C₂²×C₆₇	#536	1	16	yes	—
104	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
105	C₂²×C₆₇	C₂³×D₄	1	2	yes	—
106	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	4	yes	—
107	C₄×C₆₇	—	2	16	no	—
108	C₄×C₆₇	—	4	16	no	—
109	C₄×C₆₇	C₂²×C₂²⋊C₄	1	16	yes	—
110	C₂×#2310	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
111	C₂×C₆₇	—	2	8	no	—
112	C₂×C₆₇	#2310	1	8	yes	direct product factor
113	C₂×C₆₇	C₂³×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₆₇	C₂×#519	1	2	yes	—
115	C₂×C₆₇	C₂×#536	1	4	yes	—
116	C₂⁴×Q₈	C₂×C₆₇	1	1	yes	characteristic
117	C₂²×#117	C₂×C₆₇	1	2	yes	—
118	#2310	C₂×C₆₇	1	16	yes	direct product factor
119	C₂×#391	C₂×C₆₇	1	8	yes	complemented
120	C₂×#381	C₂×C₆₇	1	4	yes	—
121	C₆₇	C₂×#2310	1	1	yes	characteristic, direct product factor, Sylow
122	C₂³×Q₈	—	2	16	no	—
123	C₂³×Q₈	C₄×C₆₇	1	16	yes	complemented
124	C₂³×Q₈	C₂²×C₆₇	1	4	yes	—
125	C₂³×Q₈	C₂²×C₆₇	1	8	yes	—
126	C₂⁴×C₄	C₂²×C₆₇	1	2	yes	—
127	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
128	C₂×#117	—	2	16	no	—
129	C₂×#117	C₂²×C₆₇	1	16	yes	—
130	C₂²×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
131	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
132	C₂²×C₄²	C₂²×C₆₇	1	4	yes	—
133	#391	C₂²×C₆₇	1	64	yes	complemented
134	#381	C₂²×C₆₇	1	32	yes	—
135	C₂×#95	C₂²×C₆₇	1	16	yes	—
136	C₂⁵	C₂³×C₆₇	1	1	yes	characteristic
137	C₂²×Q₈	—	2	128	no	—
138	C₂²×Q₈	—	2	16	no	—
139	C₂²×Q₈	—	2	16	no	—
140	C₂²×Q₈	—	2	64	no	—
141	C₂²×Q₈	C₂×C₄×C₆₇	1	32	yes	—
142	C₂²×Q₈	C₂×C₄×C₆₇	1	64	yes	complemented
143	C₂²×Q₈	C₂³×C₆₇	1	16	yes	—
144	C₂³×C₄	—	2	16	no	—
145	C₂³×C₄	C₂×C₄×C₆₇	1	16	yes	—
146	C₂³×C₄	C₂×C₄×C₆₇	1	8	yes	—
147	C₂³×C₄	D₄×C₆₇	1	16	yes	—
148	C₂³×C₄	C₂³×C₆₇	1	8	yes	—
149	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
150	C₂³×C₄	C₂³×C₆₇	1	8	yes	—
151	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
152	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
153	#117	—	2	64	no	—
154	C₂×C₄⋊C₄	—	2	16	no	—
155	C₂×C₄⋊C₄	C₂³×C₆₇	1	16	yes	—
156	C₂×C₂²⋊C₄	—	2	16	no	—
157	C₂×C₂²⋊C₄	C₂³×C₆₇	1	16	yes	—
158	C₂×C₄²	—	2	16	no	—
159	C₂×C₄²	C₂³×C₆₇	1	16	yes	—
160	#95	C₂³×C₆₇	1	64	yes	—
161	C₂⁴	—	2	8	no	—
162	C₂⁴	C₂²×C₄×C₆₇	1	4	yes	—
163	C₂⁴	C₂×D₄×C₆₇	1	8	yes	—
164	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
165	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
166	C₂×Q₈	—	2	128	no	—
167	C₂×Q₈	—	2	256	no	—
168	C₂×Q₈	—	2	32	no	—
169	C₂×Q₈	—	2	32	no	—
170	C₂×Q₈	C₂²×C₄×C₆₇	1	64	yes	—
171	C₂²×C₄	—	2	64	no	—
172	C₂²×C₄	—	2	32	no	—
173	C₂²×C₄	—	2	16	no	—
174	C₂²×C₄	—	2	16	no	—
175	C₂²×C₄	—	2	32	no	—
176	C₂²×C₄	C₂²⋊C₄×C₆₇	1	64	yes	—
177	C₂²×C₄	C₂²×C₄×C₆₇	1	16	yes	—
178	C₂²×C₄	C₂²×C₄×C₆₇	1	32	yes	—
179	C₂²×C₄	C₂²×C₄×C₆₇	1	8	yes	—
180	C₂²×C₄	C₂²×C₄×C₆₇	1	16	yes	—
181	C₂²×C₄	C₂×D₄×C₆₇	1	8	yes	—
182	C₂²×C₄	C₂×D₄×C₆₇	1	8	yes	—
183	C₂²×C₄	C₂×D₄×C₆₇	1	32	yes	—
184	C₂²×C₄	C₂⁴×C₆₇	1	16	yes	—
185	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
186	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
187	C₄⋊C₄	—	2	32	no	—
188	C₂²⋊C₄	—	2	32	no	—
189	C₄²	—	2	32	no	—
190	C₂³	—	2	16	no	—
191	C₂³	—	2	32	no	—
192	C₂³	—	2	8	no	—
193	C₂³	C₂×C₂²⋊C₄×C₆₇	1	16	yes	—
194	C₂³	C₂³×C₄×C₆₇	1	4	yes	—
195	C₂³	C₂³×C₄×C₆₇	1	2	yes	—
196	C₂³	C₂²×D₄×C₆₇	1	8	yes	—
197	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
198	C₂³	2₋¹⁺⁴×C₆₇	1	8	yes	—
199	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
200	Q₈	—	2	128	no	—
201	C₂×C₄	—	2	64	no	—
202	C₂×C₄	—	2	32	no	—
203	C₂×C₄	—	2	16	no	—
204	C₂×C₄	—	2	16	no	—
205	C₂×C₄	—	2	32	no	—
206	C₂×C₄	—	4	32	no	—
207	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	64	yes	—
208	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	32	yes	—
209	C₂×C₄	C₂³×C₄×C₆₇	1	16	yes	—
210	C₂×C₄	C₂³×C₄×C₆₇	1	8	yes	—
211	C₂×C₄	C₂²×D₄×C₆₇	1	8	yes	—
212	C₂×C₄	C₂²×D₄×C₆₇	1	8	yes	—
213	C₂²	—	2	32	no	—
214	C₂²	—	2	8	no	—
215	C₂²	—	2	16	no	—
216	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	8	yes	—
217	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	4	yes	—
218	C₂²	#519×C₆₇	1	8	yes	—
219	C₂²	#536×C₆₇	1	16	yes	—
220	C₂²	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
221	C₂²	C₂³×D₄×C₆₇	1	2	yes	—
222	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	4	yes	—
223	C₄	—	2	16	no	—
224	C₄	—	4	16	no	—
225	C₄	C₂²×C₂²⋊C₄×C₆₇	1	16	yes	—
226	C₂	—	2	8	no	—
227	C₂	C₆₇×#2310	1	8	yes	direct product factor
228	C₂	C₂³×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
229	C₂	C₂×#519×C₆₇	1	2	yes	—
230	C₂	C₂×#536×C₆₇	1	4	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₆₇	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₆₇
Frattini, cosocle	C₂³	C₂⁵×C₆₇
prime radical	C₂⁵×C₆₇	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₆₇×#2310	C₂	direct (N × H)
4. C₂×#391×C₆₇	C₂	semidirect
7. C₂³×Q₈×C₆₇	C₄	semidirect
17. #391×C₆₇	C₂²	semidirect
26. C₂²×Q₈×C₆₇	C₂×C₄	semidirect
110. C₂×#2310	C₆₇	direct (N × H)
112. C₂×C₆₇	#2310	direct (N × H)
118. #2310	C₂×C₆₇	direct (N × H)
119. C₂×#391	C₂×C₆₇	semidirect
121. C₆₇	C₂×#2310	direct (N × H)
123. C₂³×Q₈	C₄×C₆₇	semidirect
133. #391	C₂²×C₆₇	semidirect
142. C₂²×Q₈	C₂×C₄×C₆₇	semidirect
227. C₂	C₆₇×#2310	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×Q₈×C₆₇	2
C₂²×#117×C₆₇	2
C₆₇×#2310	2
C₂×#391×C₆₇	2
C₂×#381×C₆₇	2
C₂×#2310	67

Subgroups — C2×C67×#2310 #30713293

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2310 #30713293