Subgroups — C₆₇×#14348 #30693599

Order 17152.

1262 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#384×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#382×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#376×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1442	C₂	1	8	yes	maximal
5	C₆₇×C₄⋊#95	C₂	1	4	yes	maximal
6	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
7	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
8	C₂²×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
9	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
10	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
11	#384×C₆₇	—	2	4	no	—
12	#382×C₆₇	—	2	4	no	—
13	#376×C₆₇	C₄	1	8	yes	complemented
14	C₂×#95×C₆₇	C₂²	1	4	yes	—
15	C₂×#95×C₆₇	C₂²	1	16	yes	—
16	C₂×#95×C₆₇	C₂²	1	4	yes	—
17	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
18	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
19	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
20	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
21	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
23	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
24	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
25	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
26	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
27	C₂×C₄⋊C₄×C₆₇	—	4	8	no	—
28	C₂×C₄²×C₆₇	—	2	4	no	—
29	C₂×C₄²×C₆₇	—	2	4	no	—
30	C₂×C₄²×C₆₇	C₂×C₄	1	8	yes	—
31	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
32	C₂×C₄²×C₆₇	D₄	1	4	yes	—
33	C₂×C₄²×C₆₇	Q₈	1	4	yes	—
34	#95×C₆₇	—	2	8	no	—
35	#95×C₆₇	—	4	16	no	—
36	#95×C₆₇	C₂×C₄	1	16	yes	—
37	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
38	C₂²×C₄×C₆₇	—	2	4	no	—
39	C₂²×C₄×C₆₇	—	2	4	no	—
40	C₂²×C₄×C₆₇	—	2	4	no	—
41	C₂²×C₄×C₆₇	—	2	4	no	—
42	C₂²×C₄×C₆₇	—	2	8	no	—
43	C₂²×C₄×C₆₇	—	2	8	no	—
44	C₂²×C₄×C₆₇	—	2	4	no	—
45	C₂²×C₄×C₆₇	—	2	4	no	—
46	C₂²×C₄×C₆₇	—	2	8	no	—
47	C₂²×C₄×C₆₇	—	2	8	no	—
48	C₂²×C₄×C₆₇	—	2	16	no	—
49	C₂²×C₄×C₆₇	—	2	8	no	—
50	C₂²×C₄×C₆₇	—	2	8	no	—
51	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
52	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
53	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
54	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
55	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
57	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
58	C₂²×C₄×C₆₇	#41	1	4	yes	—
59	C₂²×C₄×C₆₇	#41	1	4	yes	—
60	C₄⋊C₄×C₆₇	—	4	16	no	—
61	C₄⋊C₄×C₆₇	—	4	8	no	—
62	C₄⋊C₄×C₆₇	—	4	8	no	—
63	C₄²×C₆₇	—	2	16	no	—
64	C₄²×C₆₇	C₄⋊C₄	1	16	yes	—
65	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
66	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
67	C₂³×C₆₇	C₂²×Q₈	1	1	yes	characteristic
68	C₂³×C₆₇	C₂×#41	1	2	yes	—
69	C₂³×C₆₇	C₂×#41	1	2	yes	—
70	C₂³×C₆₇	H(5,𝔽₂)	1	4	yes	—
71	C₂³×C₆₇	2₋¹⁺⁴	1	4	yes	—
72	C₂×C₄×C₆₇	—	2	8	no	—
73	C₂×C₄×C₆₇	—	2	4	no	—
74	C₂×C₄×C₆₇	—	2	4	no	—
75	C₂×C₄×C₆₇	—	2	8	no	—
76	C₂×C₄×C₆₇	—	2	8	no	—
77	C₂×C₄×C₆₇	—	2	16	no	—
78	C₂×C₄×C₆₇	—	4	4	no	—
79	C₂×C₄×C₆₇	—	4	4	no	—
80	C₂×C₄×C₆₇	—	4	8	no	—
81	C₂×C₄×C₆₇	—	4	4	no	—
82	C₂×C₄×C₆₇	—	4	4	no	—
83	C₂×C₄×C₆₇	—	4	8	no	—
84	C₂×C₄×C₆₇	—	4	4	no	—
85	C₂×C₄×C₆₇	—	4	4	no	—
86	C₂×C₄×C₆₇	—	4	16	no	—
87	C₂×C₄×C₆₇	—	4	16	no	—
88	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	8	yes	—
89	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
90	C₂×C₄×C₆₇	#117	1	8	yes	—
91	C₂×C₄×C₆₇	#124	1	4	yes	—
92	C₂×C₄×C₆₇	#124	1	4	yes	—
93	C₂×C₄×C₆₇	#125	1	8	yes	—
94	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
95	C₂²×C₆₇	C₂×#117	1	2	yes	—
96	C₂²×C₆₇	#520	1	4	yes	—
97	C₂²×C₆₇	C₂×#124	1	1	yes	characteristic
98	C₂²×C₆₇	C₂×#124	1	1	yes	characteristic
99	C₂²×C₆₇	C₂×#125	1	2	yes	—
100	C₂²×C₆₇	#535	1	2	yes	—
101	C₂²×C₆₇	#536	1	2	yes	—
102	C₂²×C₆₇	#540	1	4	yes	—
103	C₂²×C₆₇	#541	1	4	yes	—
104	C₂²×C₆₇	#542	1	8	yes	—
105	C₂²×C₆₇	#544	1	4	yes	—
106	C₄×C₆₇	—	2	8	no	—
107	C₄×C₆₇	—	4	8	no	—
108	C₄×C₆₇	—	8	4	no	—
109	C₄×C₆₇	—	8	4	no	—
110	C₄×C₆₇	#382	1	8	yes	—
111	#14348	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
112	C₂×C₆₇	C₂×#382	1	1	yes	characteristic
113	C₂×C₆₇	#2314	1	4	yes	—
114	C₂×C₆₇	#2317	1	2	yes	—
115	C₂×C₆₇	#2640	1	2	yes	—
116	C₂×C₆₇	#2641	1	1	yes	characteristic
117	C₂×C₆₇	#2642	1	1	yes	characteristic
118	C₂×C₆₇	#2647	1	4	yes	—
119	C₂×#384	C₂×C₆₇	1	1	yes	characteristic
120	C₂×#382	C₂×C₆₇	1	1	yes	characteristic
121	C₂×#376	C₂×C₆₇	1	1	yes	characteristic
122	#1442	C₂×C₆₇	1	8	yes	—
123	C₄⋊#95	C₂×C₆₇	1	4	yes	—
124	C₆₇	#14348	1	1	yes	characteristic, direct product factor, Sylow
125	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
126	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
127	C₂²×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
128	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
129	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
130	#384	—	2	4	no	—
131	#382	—	2	4	no	—
132	#376	C₄×C₆₇	1	8	yes	complemented
133	C₂×#95	C₂²×C₆₇	1	4	yes	—
134	C₂×#95	C₂²×C₆₇	1	16	yes	—
135	C₂×#95	C₂²×C₆₇	1	4	yes	—
136	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
137	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
138	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
139	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
140	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
141	C₂×C₄⋊C₄	—	2	4	no	—
142	C₂×C₄⋊C₄	—	2	8	no	—
143	C₂×C₄⋊C₄	—	2	4	no	—
144	C₂×C₄⋊C₄	—	2	8	no	—
145	C₂×C₄⋊C₄	—	2	8	no	—
146	C₂×C₄⋊C₄	—	4	8	no	—
147	C₂×C₄²	—	2	4	no	—
148	C₂×C₄²	—	2	4	no	—
149	C₂×C₄²	C₂×C₄×C₆₇	1	8	yes	—
150	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
151	C₂×C₄²	D₄×C₆₇	1	4	yes	—
152	C₂×C₄²	Q₈×C₆₇	1	4	yes	—
153	#95	—	2	8	no	—
154	#95	—	4	16	no	—
155	#95	C₂×C₄×C₆₇	1	16	yes	—
156	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
157	C₂²×C₄	—	2	4	no	—
158	C₂²×C₄	—	2	4	no	—
159	C₂²×C₄	—	2	4	no	—
160	C₂²×C₄	—	2	4	no	—
161	C₂²×C₄	—	2	8	no	—
162	C₂²×C₄	—	2	8	no	—
163	C₂²×C₄	—	2	4	no	—
164	C₂²×C₄	—	2	4	no	—
165	C₂²×C₄	—	2	8	no	—
166	C₂²×C₄	—	2	8	no	—
167	C₂²×C₄	—	2	8	no	—
168	C₂²×C₄	—	2	16	no	—
169	C₂²×C₄	—	2	8	no	—
170	C₂²×C₄	C₂²×C₄×C₆₇	1	8	yes	—
171	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
172	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
173	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
174	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
175	C₂²×C₄	C₂×Q₈×C₆₇	1	4	yes	—
176	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
177	C₂²×C₄	#41×C₆₇	1	4	yes	—
178	C₂²×C₄	#41×C₆₇	1	4	yes	—
179	C₄⋊C₄	—	4	16	no	—
180	C₄⋊C₄	—	4	8	no	—
181	C₄⋊C₄	—	4	8	no	—
182	C₄²	—	2	16	no	—
183	C₄²	C₄⋊C₄×C₆₇	1	16	yes	—
184	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
185	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
186	C₂³	C₂²×Q₈×C₆₇	1	1	yes	characteristic
187	C₂³	C₂×#41×C₆₇	1	2	yes	—
188	C₂³	C₂×#41×C₆₇	1	2	yes	—
189	C₂³	H(5,𝔽₂)×C₆₇	1	4	yes	—
190	C₂³	2₋¹⁺⁴×C₆₇	1	4	yes	—
191	C₂×C₄	—	2	8	no	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	2	4	no	—
194	C₂×C₄	—	2	8	no	—
195	C₂×C₄	—	2	8	no	—
196	C₂×C₄	—	2	16	no	—
197	C₂×C₄	—	4	4	no	—
198	C₂×C₄	—	4	4	no	—
199	C₂×C₄	—	4	8	no	—
200	C₂×C₄	—	4	4	no	—
201	C₂×C₄	—	4	4	no	—
202	C₂×C₄	—	4	8	no	—
203	C₂×C₄	—	4	4	no	—
204	C₂×C₄	—	4	4	no	—
205	C₂×C₄	—	4	16	no	—
206	C₂×C₄	—	4	16	no	—
207	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	8	yes	—
208	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
209	C₂×C₄	#117×C₆₇	1	8	yes	—
210	C₂×C₄	#124×C₆₇	1	4	yes	—
211	C₂×C₄	#124×C₆₇	1	4	yes	—
212	C₂×C₄	#125×C₆₇	1	8	yes	—
213	C₂²	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
214	C₂²	C₂×#117×C₆₇	1	2	yes	—
215	C₂²	#520×C₆₇	1	4	yes	—
216	C₂²	C₂×#124×C₆₇	1	1	yes	characteristic
217	C₂²	C₂×#124×C₆₇	1	1	yes	characteristic
218	C₂²	C₂×#125×C₆₇	1	2	yes	—
219	C₂²	#535×C₆₇	1	2	yes	—
220	C₂²	#536×C₆₇	1	2	yes	—
221	C₂²	#540×C₆₇	1	4	yes	—
222	C₂²	#541×C₆₇	1	4	yes	—
223	C₂²	#542×C₆₇	1	8	yes	—
224	C₂²	#544×C₆₇	1	4	yes	—
225	C₄	—	2	8	no	—
226	C₄	—	4	8	no	—
227	C₄	—	8	4	no	—
228	C₄	—	8	4	no	—
229	C₄	#382×C₆₇	1	8	yes	—
230	C₂	C₂×#382×C₆₇	1	1	yes	characteristic
231	C₂	C₆₇×#2314	1	4	yes	—
232	C₂	C₆₇×#2317	1	2	yes	—
233	C₂	C₆₇×#2640	1	2	yes	—
234	C₂	C₆₇×#2641	1	1	yes	characteristic
235	C₂	C₆₇×#2642	1	1	yes	characteristic
236	C₂	C₆₇×#2647	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
13. #376×C₆₇	C₄	semidirect
111. #14348	C₆₇	direct (N × H)
124. C₆₇	#14348	direct (N × H)
132. #376	C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#384×C₆₇	2
C₂×#382×C₆₇	2
C₂×#376×C₆₇	2
C₆₇×#1442	2
C₆₇×C₄⋊#95	2
#14348	67

Subgroups — C67×#14348 #30693599

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#14348 #30693599