Subgroups — C₆₇×#34171 #30713422

Order 17152.

7734 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#518×C₆₇	C₂	1	2	yes	maximal, complemented
3	D₄×C₂²⋊C₄×C₆₇	C₂	1	16	yes	maximal, complemented
4	C₆₇×#2309	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2308	C₂	1	4	yes	maximal, complemented
6	C₆₇×#2280	C₂	1	2	yes	maximal, complemented
7	C₂×#379×C₆₇	C₂	1	4	yes	maximal, complemented
8	C₂⁶×C₆₇	C₂²	1	2	yes	—
9	C₂³×D₄×C₆₇	—	2	16	no	—
10	C₂³×D₄×C₆₇	C₄	1	16	yes	complemented
11	C₂³×D₄×C₆₇	C₂²	1	2	yes	—
12	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
13	C₂³×D₄×C₆₇	C₂²	1	8	yes	—
14	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
15	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
16	#518×C₆₇	—	2	16	no	—
17	C₂×C₄×D₄×C₆₇	C₂²	1	16	yes	complemented
18	C₂×#117×C₆₇	C₂²	1	4	yes	complemented
19	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	8	yes	complemented
20	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	16	yes	complemented
21	#390×C₆₇	C₂²	1	16	yes	complemented
22	#386×C₆₇	C₂²	1	32	yes	complemented
23	#381×C₆₇	C₂²	1	4	yes	complemented
24	#380×C₆₇	C₂²	1	8	yes	complemented
25	#379×C₆₇	C₂²	1	4	yes	—
26	#379×C₆₇	C₂²	1	8	yes	complemented
27	#379×C₆₇	C₂²	1	16	yes	complemented
28	C₄×C₂²⋊C₄×C₆₇	C₂²	1	8	yes	complemented
29	C₂⁵×C₆₇	—	2	16	no	—
30	C₂⁵×C₆₇	—	4	16	no	—
31	C₂⁵×C₆₇	C₂×C₄	1	16	yes	complemented
32	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
33	C₂⁵×C₆₇	C₂³	1	8	yes	—
34	C₂⁵×C₆₇	C₂³	1	2	yes	—
35	C₂⁵×C₆₇	C₂³	1	2	yes	—
36	C₂²×D₄×C₆₇	—	2	16	no	—
37	C₂²×D₄×C₆₇	—	2	128	no	—
38	C₂²×D₄×C₆₇	—	2	64	no	—
39	C₂²×D₄×C₆₇	—	2	8	no	—
40	C₂²×D₄×C₆₇	C₂×C₄	1	8	yes	—
41	C₂²×D₄×C₆₇	C₂×C₄	1	16	yes	complemented
42	C₂²×D₄×C₆₇	C₂×C₄	1	64	yes	complemented
43	C₂²×D₄×C₆₇	C₂×C₄	1	8	yes	complemented
44	C₂²×D₄×C₆₇	D₄	1	16	yes	—
45	C₂²×D₄×C₆₇	C₂³	1	8	yes	—
46	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
47	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
48	C₂³×C₄×C₆₇	—	2	8	no	—
49	C₂³×C₄×C₆₇	C₂×C₄	1	8	yes	complemented
50	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
51	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
52	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
53	C₂³×C₄×C₆₇	C₂³	1	16	yes	complemented
54	C₄×D₄×C₆₇	—	2	64	no	—
55	#117×C₆₇	—	2	16	no	—
56	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	complemented
57	C₂×C₂²⋊C₄×C₆₇	—	2	32	no	—
58	C₂×C₂²⋊C₄×C₆₇	—	4	32	no	—
59	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
60	C₂×C₂²⋊C₄×C₆₇	C₂³	1	16	yes	—
61	C₂×C₂²⋊C₄×C₆₇	C₂³	1	16	yes	complemented
62	C₂×C₂²⋊C₄×C₆₇	C₂³	1	32	yes	complemented
63	C₂×C₄²×C₆₇	C₂³	1	8	yes	complemented
64	#95×C₆₇	C₂³	1	16	yes	complemented
65	C₂⁴×C₆₇	—	2	4	no	—
66	C₂⁴×C₆₇	—	2	8	no	—
67	C₂⁴×C₆₇	—	2	64	no	—
68	C₂⁴×C₆₇	—	2	32	no	—
69	C₂⁴×C₆₇	—	2	8	no	—
70	C₂⁴×C₆₇	—	2	32	no	—
71	C₂⁴×C₆₇	—	4	64	no	—
72	C₂⁴×C₆₇	—	4	8	no	—
73	C₂⁴×C₆₇	—	4	16	no	—
74	C₂⁴×C₆₇	—	4	8	no	—
75	C₂⁴×C₆₇	—	4	128	no	—
76	C₂⁴×C₆₇	C₂²×C₄	1	4	yes	—
77	C₂⁴×C₆₇	C₂²×C₄	1	32	yes	—
78	C₂⁴×C₆₇	C₂²×C₄	1	8	yes	—
79	C₂⁴×C₆₇	C₂²×C₄	1	8	yes	—
80	C₂⁴×C₆₇	C₂×D₄	1	16	yes	—
81	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
82	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
83	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
84	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
85	C₂×D₄×C₆₇	—	2	64	no	—
86	C₂×D₄×C₆₇	—	2	32	no	—
87	C₂×D₄×C₆₇	—	2	256	no	—
88	C₂×D₄×C₆₇	—	2	32	no	—
89	C₂×D₄×C₆₇	—	2	16	no	—
90	C₂×D₄×C₆₇	C₂²⋊C₄	1	64	yes	complemented
91	C₂×D₄×C₆₇	C₂²×C₄	1	32	yes	—
92	C₂×D₄×C₆₇	C₂²×C₄	1	16	yes	—
93	C₂×D₄×C₆₇	C₂²×C₄	1	16	yes	—
94	C₂×D₄×C₆₇	C₂×D₄	1	32	yes	—
95	C₂²×C₄×C₆₇	—	2	4	no	—
96	C₂²×C₄×C₆₇	—	2	32	no	—
97	C₂²×C₄×C₆₇	—	2	16	no	—
98	C₂²×C₄×C₆₇	—	2	32	no	—
99	C₂²×C₄×C₆₇	—	4	32	no	—
100	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
101	C₂²×C₄×C₆₇	C₂²×C₄	1	16	yes	—
102	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
103	C₂²×C₄×C₆₇	C₂×D₄	1	8	yes	—
104	C₂²×C₄×C₆₇	C₂⁴	1	16	yes	—
105	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
106	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
107	C₄⋊C₄×C₆₇	—	2	16	no	—
108	C₂²⋊C₄×C₆₇	—	2	16	no	—
109	C₂²⋊C₄×C₆₇	—	2	32	no	—
110	C₂²⋊C₄×C₆₇	—	2	32	no	—
111	C₂²⋊C₄×C₆₇	—	4	32	no	—
112	C₄²×C₆₇	—	2	16	no	—
113	C₂³×C₆₇	—	2	32	no	—
114	C₂³×C₆₇	—	2	8	no	—
115	C₂³×C₆₇	—	2	16	no	—
116	C₂³×C₆₇	—	2	4	no	—
117	C₂³×C₆₇	—	2	16	no	—
118	C₂³×C₆₇	—	2	4	no	—
119	C₂³×C₆₇	—	2	4	no	—
120	C₂³×C₆₇	—	2	64	no	—
121	C₂³×C₆₇	—	2	16	no	—
122	C₂³×C₆₇	—	2	64	no	—
123	C₂³×C₆₇	—	2	8	no	—
124	C₂³×C₆₇	—	2	16	no	—
125	C₂³×C₆₇	—	4	16	no	—
126	C₂³×C₆₇	—	4	32	no	—
127	C₂³×C₆₇	—	4	16	no	—
128	C₂³×C₆₇	—	4	32	no	—
129	C₂³×C₆₇	—	4	128	no	—
130	C₂³×C₆₇	—	4	256	no	—
131	C₂³×C₆₇	—	4	32	no	—
132	C₂³×C₆₇	C₂×C₂²⋊C₄	1	32	yes	complemented
133	C₂³×C₆₇	C₂³×C₄	1	8	yes	—
134	C₂³×C₆₇	C₂³×C₄	1	2	yes	—
135	C₂³×C₆₇	C₂³×C₄	1	8	yes	—
136	C₂³×C₆₇	C₂³×C₄	1	4	yes	—
137	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
138	C₂³×C₆₇	C₂²×D₄	1	16	yes	—
139	C₂³×C₆₇	H(5,𝔽₂)	1	8	yes	—
140	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
141	D₄×C₆₇	—	2	64	no	—
142	D₄×C₆₇	—	2	32	no	—
143	D₄×C₆₇	C₂×C₂²⋊C₄	1	64	yes	complemented
144	C₂×C₄×C₆₇	—	2	8	no	—
145	C₂×C₄×C₆₇	—	2	4	no	—
146	C₂×C₄×C₆₇	—	2	32	no	—
147	C₂×C₄×C₆₇	—	2	16	no	—
148	C₂×C₄×C₆₇	—	2	32	no	—
149	C₂×C₄×C₆₇	—	8	32	no	—
150	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	16	yes	complemented
151	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
152	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
153	C₂×C₄×C₆₇	C₂²×D₄	1	8	yes	—
154	C₂²×C₆₇	—	2	32	no	—
155	C₂²×C₆₇	—	2	4	no	—
156	C₂²×C₆₇	—	2	4	no	—
157	C₂²×C₆₇	—	2	4	no	—
158	C₂²×C₆₇	—	2	16	no	—
159	C₂²×C₆₇	—	2	32	no	—
160	C₂²×C₆₇	—	2	8	no	—
161	C₂²×C₆₇	—	2	16	no	—
162	C₂²×C₆₇	—	2	8	no	—
163	C₂²×C₆₇	—	2	16	no	—
164	C₂²×C₆₇	—	4	32	no	—
165	C₂²×C₆₇	—	4	128	no	—
166	C₂²×C₆₇	—	4	32	no	—
167	C₂²×C₆₇	—	4	16	no	—
168	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	4	yes	—
169	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	16	yes	—
170	C₂²×C₆₇	#518	1	8	yes	complemented
171	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
172	C₂²×C₆₇	C₂³×D₄	1	2	yes	—
173	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	4	yes	—
174	C₄×C₆₇	—	2	4	no	—
175	C₄×C₆₇	—	4	16	no	—
176	C₄×C₆₇	C₂²×C₂²⋊C₄	1	8	yes	—
177	#34171	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
178	C₂×C₆₇	—	2	4	no	—
179	C₂×C₆₇	—	2	8	no	—
180	C₂×C₆₇	—	2	16	no	—
181	C₂×C₆₇	—	4	8	no	—
182	C₂×C₆₇	C₂³×C₂²⋊C₄	1	1	yes	characteristic
183	C₂×C₆₇	C₂×#518	1	2	yes	—
184	C₂×C₆₇	#3484	1	4	yes	—
185	C₂⁴×D₄	C₂×C₆₇	1	1	yes	characteristic
186	C₂×#518	C₂×C₆₇	1	2	yes	complemented
187	D₄×C₂²⋊C₄	C₂×C₆₇	1	16	yes	complemented
188	#2309	C₂×C₆₇	1	2	yes	complemented
189	#2308	C₂×C₆₇	1	4	yes	complemented
190	#2280	C₂×C₆₇	1	2	yes	complemented
191	C₂×#379	C₂×C₆₇	1	4	yes	complemented
192	C₆₇	#34171	1	1	yes	characteristic, direct product factor, Sylow
193	C₂⁶	C₂²×C₆₇	1	2	yes	—
194	C₂³×D₄	—	2	16	no	—
195	C₂³×D₄	C₄×C₆₇	1	16	yes	complemented
196	C₂³×D₄	C₂²×C₆₇	1	2	yes	—
197	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
198	C₂³×D₄	C₂²×C₆₇	1	8	yes	—
199	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
200	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
201	#518	—	2	16	no	—
202	C₂×C₄×D₄	C₂²×C₆₇	1	16	yes	complemented
203	C₂×#117	C₂²×C₆₇	1	4	yes	complemented
204	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	8	yes	complemented
205	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	16	yes	complemented
206	#390	C₂²×C₆₇	1	16	yes	complemented
207	#386	C₂²×C₆₇	1	32	yes	complemented
208	#381	C₂²×C₆₇	1	4	yes	complemented
209	#380	C₂²×C₆₇	1	8	yes	complemented
210	#379	C₂²×C₆₇	1	4	yes	—
211	#379	C₂²×C₆₇	1	8	yes	complemented
212	#379	C₂²×C₆₇	1	16	yes	complemented
213	C₄×C₂²⋊C₄	C₂²×C₆₇	1	8	yes	complemented
214	C₂⁵	—	2	16	no	—
215	C₂⁵	—	4	16	no	—
216	C₂⁵	C₂×C₄×C₆₇	1	16	yes	complemented
217	C₂⁵	C₂³×C₆₇	1	1	yes	characteristic
218	C₂⁵	C₂³×C₆₇	1	8	yes	—
219	C₂⁵	C₂³×C₆₇	1	2	yes	—
220	C₂⁵	C₂³×C₆₇	1	2	yes	—
221	C₂²×D₄	—	2	16	no	—
222	C₂²×D₄	—	2	128	no	—
223	C₂²×D₄	—	2	64	no	—
224	C₂²×D₄	—	2	8	no	—
225	C₂²×D₄	C₂×C₄×C₆₇	1	8	yes	—
226	C₂²×D₄	C₂×C₄×C₆₇	1	16	yes	complemented
227	C₂²×D₄	C₂×C₄×C₆₇	1	64	yes	complemented
228	C₂²×D₄	C₂×C₄×C₆₇	1	8	yes	complemented
229	C₂²×D₄	D₄×C₆₇	1	16	yes	—
230	C₂²×D₄	C₂³×C₆₇	1	8	yes	—
231	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
232	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
233	C₂³×C₄	—	2	8	no	—
234	C₂³×C₄	C₂×C₄×C₆₇	1	8	yes	complemented
235	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
236	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
237	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
238	C₂³×C₄	C₂³×C₆₇	1	16	yes	complemented
239	C₄×D₄	—	2	64	no	—
240	#117	—	2	16	no	—
241	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	complemented
242	C₂×C₂²⋊C₄	—	2	32	no	—
243	C₂×C₂²⋊C₄	—	4	32	no	—
244	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
245	C₂×C₂²⋊C₄	C₂³×C₆₇	1	16	yes	—
246	C₂×C₂²⋊C₄	C₂³×C₆₇	1	16	yes	complemented
247	C₂×C₂²⋊C₄	C₂³×C₆₇	1	32	yes	complemented
248	C₂×C₄²	C₂³×C₆₇	1	8	yes	complemented
249	#95	C₂³×C₆₇	1	16	yes	complemented
250	C₂⁴	—	2	4	no	—
251	C₂⁴	—	2	8	no	—
252	C₂⁴	—	2	64	no	—
253	C₂⁴	—	2	32	no	—
254	C₂⁴	—	2	8	no	—
255	C₂⁴	—	2	32	no	—
256	C₂⁴	—	4	64	no	—
257	C₂⁴	—	4	8	no	—
258	C₂⁴	—	4	16	no	—
259	C₂⁴	—	4	8	no	—
260	C₂⁴	—	4	128	no	—
261	C₂⁴	C₂²×C₄×C₆₇	1	4	yes	—
262	C₂⁴	C₂²×C₄×C₆₇	1	32	yes	—
263	C₂⁴	C₂²×C₄×C₆₇	1	8	yes	—
264	C₂⁴	C₂²×C₄×C₆₇	1	8	yes	—
265	C₂⁴	C₂×D₄×C₆₇	1	16	yes	—
266	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
267	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
268	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
269	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
270	C₂×D₄	—	2	64	no	—
271	C₂×D₄	—	2	32	no	—
272	C₂×D₄	—	2	256	no	—
273	C₂×D₄	—	2	32	no	—
274	C₂×D₄	—	2	16	no	—
275	C₂×D₄	C₂²⋊C₄×C₆₇	1	64	yes	complemented
276	C₂×D₄	C₂²×C₄×C₆₇	1	32	yes	—
277	C₂×D₄	C₂²×C₄×C₆₇	1	16	yes	—
278	C₂×D₄	C₂²×C₄×C₆₇	1	16	yes	—
279	C₂×D₄	C₂×D₄×C₆₇	1	32	yes	—
280	C₂²×C₄	—	2	4	no	—
281	C₂²×C₄	—	2	32	no	—
282	C₂²×C₄	—	2	16	no	—
283	C₂²×C₄	—	2	32	no	—
284	C₂²×C₄	—	4	32	no	—
285	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
286	C₂²×C₄	C₂²×C₄×C₆₇	1	16	yes	—
287	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
288	C₂²×C₄	C₂×D₄×C₆₇	1	8	yes	—
289	C₂²×C₄	C₂⁴×C₆₇	1	16	yes	—
290	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
291	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
292	C₄⋊C₄	—	2	16	no	—
293	C₂²⋊C₄	—	2	16	no	—
294	C₂²⋊C₄	—	2	32	no	—
295	C₂²⋊C₄	—	2	32	no	—
296	C₂²⋊C₄	—	4	32	no	—
297	C₄²	—	2	16	no	—
298	C₂³	—	2	32	no	—
299	C₂³	—	2	8	no	—
300	C₂³	—	2	16	no	—
301	C₂³	—	2	4	no	—
302	C₂³	—	2	16	no	—
303	C₂³	—	2	4	no	—
304	C₂³	—	2	4	no	—
305	C₂³	—	2	64	no	—
306	C₂³	—	2	16	no	—
307	C₂³	—	2	64	no	—
308	C₂³	—	2	8	no	—
309	C₂³	—	2	16	no	—
310	C₂³	—	4	16	no	—
311	C₂³	—	4	32	no	—
312	C₂³	—	4	16	no	—
313	C₂³	—	4	32	no	—
314	C₂³	—	4	128	no	—
315	C₂³	—	4	256	no	—
316	C₂³	—	4	32	no	—
317	C₂³	C₂×C₂²⋊C₄×C₆₇	1	32	yes	complemented
318	C₂³	C₂³×C₄×C₆₇	1	8	yes	—
319	C₂³	C₂³×C₄×C₆₇	1	2	yes	—
320	C₂³	C₂³×C₄×C₆₇	1	8	yes	—
321	C₂³	C₂³×C₄×C₆₇	1	4	yes	—
322	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
323	C₂³	C₂²×D₄×C₆₇	1	16	yes	—
324	C₂³	H(5,𝔽₂)×C₆₇	1	8	yes	—
325	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
326	D₄	—	2	64	no	—
327	D₄	—	2	32	no	—
328	D₄	C₂×C₂²⋊C₄×C₆₇	1	64	yes	complemented
329	C₂×C₄	—	2	8	no	—
330	C₂×C₄	—	2	4	no	—
331	C₂×C₄	—	2	32	no	—
332	C₂×C₄	—	2	16	no	—
333	C₂×C₄	—	2	32	no	—
334	C₂×C₄	—	8	32	no	—
335	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	16	yes	complemented
336	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
337	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
338	C₂×C₄	C₂²×D₄×C₆₇	1	8	yes	—
339	C₂²	—	2	32	no	—
340	C₂²	—	2	4	no	—
341	C₂²	—	2	4	no	—
342	C₂²	—	2	4	no	—
343	C₂²	—	2	16	no	—
344	C₂²	—	2	32	no	—
345	C₂²	—	2	8	no	—
346	C₂²	—	2	16	no	—
347	C₂²	—	2	8	no	—
348	C₂²	—	2	16	no	—
349	C₂²	—	4	32	no	—
350	C₂²	—	4	128	no	—
351	C₂²	—	4	32	no	—
352	C₂²	—	4	16	no	—
353	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	4	yes	—
354	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	16	yes	—
355	C₂²	#518×C₆₇	1	8	yes	complemented
356	C₂²	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
357	C₂²	C₂³×D₄×C₆₇	1	2	yes	—
358	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	4	yes	—
359	C₄	—	2	4	no	—
360	C₄	—	4	16	no	—
361	C₄	C₂²×C₂²⋊C₄×C₆₇	1	8	yes	—
362	C₂	—	2	4	no	—
363	C₂	—	2	8	no	—
364	C₂	—	2	16	no	—
365	C₂	—	4	8	no	—
366	C₂	C₂³×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
367	C₂	C₂×#518×C₆₇	1	2	yes	—
368	C₂	C₆₇×#3484	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₆₇
Frattini, cosocle	C₂³	C₂⁵×C₆₇
prime radical	C₂⁴×D₄×C₆₇	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#518×C₆₇	C₂	semidirect
3. D₄×C₂²⋊C₄×C₆₇	C₂	semidirect
4. C₆₇×#2309	C₂	semidirect
5. C₆₇×#2308	C₂	semidirect
6. C₆₇×#2280	C₂	semidirect
7. C₂×#379×C₆₇	C₂	semidirect
10. C₂³×D₄×C₆₇	C₄	semidirect
17. C₂×C₄×D₄×C₆₇	C₂²	semidirect
18. C₂×#117×C₆₇	C₂²	semidirect
19. C₂²×C₂²⋊C₄×C₆₇	C₂²	semidirect
20. C₂²×C₂²⋊C₄×C₆₇	C₂²	semidirect
21. #390×C₆₇	C₂²	semidirect
22. #386×C₆₇	C₂²	semidirect
23. #381×C₆₇	C₂²	semidirect
24. #380×C₆₇	C₂²	semidirect
26. #379×C₆₇	C₂²	semidirect
27. #379×C₆₇	C₂²	semidirect
28. C₄×C₂²⋊C₄×C₆₇	C₂²	semidirect
31. C₂⁵×C₆₇	C₂×C₄	semidirect
41. C₂²×D₄×C₆₇	C₂×C₄	semidirect
42. C₂²×D₄×C₆₇	C₂×C₄	semidirect
43. C₂²×D₄×C₆₇	C₂×C₄	semidirect
49. C₂³×C₄×C₆₇	C₂×C₄	semidirect
53. C₂³×C₄×C₆₇	C₂³	semidirect
56. C₂×C₄⋊C₄×C₆₇	C₂³	semidirect
61. C₂×C₂²⋊C₄×C₆₇	C₂³	semidirect
62. C₂×C₂²⋊C₄×C₆₇	C₂³	semidirect
63. C₂×C₄²×C₆₇	C₂³	semidirect
64. #95×C₆₇	C₂³	semidirect
90. C₂×D₄×C₆₇	C₂²⋊C₄	semidirect
132. C₂³×C₆₇	C₂×C₂²⋊C₄	semidirect
143. D₄×C₆₇	C₂×C₂²⋊C₄	semidirect
150. C₂×C₄×C₆₇	C₂×C₂²⋊C₄	semidirect
170. C₂²×C₆₇	#518	semidirect
177. #34171	C₆₇	direct (N × H)
186. C₂×#518	C₂×C₆₇	semidirect
187. D₄×C₂²⋊C₄	C₂×C₆₇	semidirect
188. #2309	C₂×C₆₇	semidirect
189. #2308	C₂×C₆₇	semidirect
190. #2280	C₂×C₆₇	semidirect
191. C₂×#379	C₂×C₆₇	semidirect
192. C₆₇	#34171	direct (N × H)
195. C₂³×D₄	C₄×C₆₇	semidirect
202. C₂×C₄×D₄	C₂²×C₆₇	semidirect
203. C₂×#117	C₂²×C₆₇	semidirect
204. C₂²×C₂²⋊C₄	C₂²×C₆₇	semidirect
205. C₂²×C₂²⋊C₄	C₂²×C₆₇	semidirect
206. #390	C₂²×C₆₇	semidirect
207. #386	C₂²×C₆₇	semidirect
208. #381	C₂²×C₆₇	semidirect
209. #380	C₂²×C₆₇	semidirect
211. #379	C₂²×C₆₇	semidirect
212. #379	C₂²×C₆₇	semidirect
213. C₄×C₂²⋊C₄	C₂²×C₆₇	semidirect
216. C₂⁵	C₂×C₄×C₆₇	semidirect
226. C₂²×D₄	C₂×C₄×C₆₇	semidirect
227. C₂²×D₄	C₂×C₄×C₆₇	semidirect
228. C₂²×D₄	C₂×C₄×C₆₇	semidirect
234. C₂³×C₄	C₂×C₄×C₆₇	semidirect
238. C₂³×C₄	C₂³×C₆₇	semidirect
241. C₂×C₄⋊C₄	C₂³×C₆₇	semidirect
246. C₂×C₂²⋊C₄	C₂³×C₆₇	semidirect
247. C₂×C₂²⋊C₄	C₂³×C₆₇	semidirect
248. C₂×C₄²	C₂³×C₆₇	semidirect
249. #95	C₂³×C₆₇	semidirect
275. C₂×D₄	C₂²⋊C₄×C₆₇	semidirect
317. C₂³	C₂×C₂²⋊C₄×C₆₇	semidirect
328. D₄	C₂×C₂²⋊C₄×C₆₇	semidirect
335. C₂×C₄	C₂×C₂²⋊C₄×C₆₇	semidirect
355. C₂²	#518×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂⁴×D₄×C₆₇	2
C₂×#518×C₆₇	2
D₄×C₂²⋊C₄×C₆₇	2
C₆₇×#2309	2
C₆₇×#2308	2
C₆₇×#2280	2
C₂×#379×C₆₇	2
#34171	67

Subgroups — C67×#34171 #30713422

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#34171 #30713422