Subgroups — C₂×C₆₇×#3396 #30740643

Order 17152.

2086 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#559×C₆₇	C₂	1	2	yes	maximal, complemented
2	C₆₇×#3396	C₂	1	16	yes	maximal, direct product factor
3	C₂×#495×C₆₇	C₂	1	2	yes	maximal, complemented
4	C₂×#493×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#460×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₂×#463×C₆₇	C₂	1	4	yes	maximal, complemented
7	C₂×C₈×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×Q₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₄×D₈×C₆₇	C₂	1	2	yes	maximal, complemented
10	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
11	C₂²×SD₁₆×C₆₇	C₂²	1	4	yes	—
12	C₂²×D₈×C₆₇	C₂²	1	2	yes	—
13	C₂²×D₈×C₆₇	C₂²	1	2	yes	—
14	#559×C₆₇	—	2	8	no	—
15	#559×C₆₇	C₂²	1	16	yes	complemented
16	C₂×#127×C₆₇	C₂²	1	2	yes	—
17	C₂×#124×C₆₇	—	2	2	no	—
18	C₂×#124×C₆₇	C₂²	1	4	yes	—
19	C₂×#123×C₆₇	—	2	4	no	—
20	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
21	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×C₄×D₄×C₆₇	—	2	2	no	—
23	C₂×C₄×D₄×C₆₇	C₂²	1	4	yes	—
24	#495×C₆₇	C₂²	1	16	yes	complemented
25	#493×C₆₇	C₂²	1	8	yes	—
26	#463×C₆₇	C₂²	1	32	yes	complemented
27	#460×C₆₇	C₂²	1	16	yes	complemented
28	C₈×Q₈×C₆₇	C₂²	1	8	yes	complemented
29	C₄×Q₁₆×C₆₇	C₂²	1	8	yes	complemented
30	C₄×D₈×C₆₇	C₂²	1	16	yes	complemented
31	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
32	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
33	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
34	C₂×#103×C₆₇	C₂²	1	2	yes	—
35	C₂×#102×C₆₇	C₂²	1	4	yes	—
36	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
37	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
38	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
39	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
40	C₂²×D₄×C₆₇	—	2	2	no	—
41	C₂²×D₄×C₆₇	—	2	2	no	—
42	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
43	C₂³×C₄×C₆₇	—	2	4	no	—
44	C₂×Q₁₆×C₆₇	—	2	4	no	—
45	C₂×Q₁₆×C₆₇	C₂³	1	4	yes	—
46	C₂×SD₁₆×C₆₇	—	4	8	no	—
47	C₂×SD₁₆×C₆₇	C₂³	1	16	yes	—
48	C₂×D₈×C₆₇	—	2	8	no	—
49	C₂×D₈×C₆₇	—	4	4	no	—
50	C₂×D₈×C₆₇	C₂³	1	8	yes	—
51	C₂×D₈×C₆₇	C₂³	1	8	yes	—
52	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
54	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
55	#127×C₆₇	—	2	4	no	—
56	#127×C₆₇	C₂³	1	8	yes	—
57	#124×C₆₇	—	2	16	no	—
58	#124×C₆₇	—	2	8	no	—
59	#124×C₆₇	C₂³	1	16	yes	—
60	#123×C₆₇	—	2	32	no	—
61	C₄×Q₈×C₆₇	—	2	4	no	—
62	C₄×Q₈×C₆₇	D₄	1	4	yes	—
63	C₄×Q₈×C₆₇	C₂³	1	8	yes	—
64	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
65	C₄×D₄×C₆₇	—	2	16	no	—
66	C₄×D₄×C₆₇	—	2	8	no	—
67	C₄×D₄×C₆₇	C₂³	1	16	yes	—
68	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
69	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
70	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
71	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
72	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
73	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
74	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
75	C₂×C₄²×C₆₇	—	2	2	no	—
76	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
77	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
78	#107×C₆₇	C₂³	1	4	yes	—
79	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
80	C₄⋊C₈×C₆₇	C₂³	1	8	yes	—
81	#103×C₆₇	C₂³	1	8	yes	—
82	#102×C₆₇	C₂³	1	16	yes	—
83	C₄×C₈×C₆₇	C₂³	1	4	yes	—
84	C₄×C₈×C₆₇	C₂³	1	8	yes	—
85	C₂⁴×C₆₇	—	2	4	no	—
86	C₂×Q₈×C₆₇	—	2	4	no	—
87	C₂×Q₈×C₆₇	—	2	2	no	—
88	C₂×Q₈×C₆₇	—	2	4	no	—
89	C₂×Q₈×C₆₇	—	2	4	no	—
90	C₂×Q₈×C₆₇	C₂×D₄	1	4	yes	—
91	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
92	C₂×D₄×C₆₇	—	2	4	no	—
93	C₂×D₄×C₆₇	—	2	8	no	—
94	C₂×D₄×C₆₇	—	2	8	no	—
95	C₂×D₄×C₆₇	—	2	8	no	—
96	C₂×D₄×C₆₇	—	2	8	no	—
97	C₂×D₄×C₆₇	—	4	8	no	—
98	C₂×D₄×C₆₇	—	8	4	no	—
99	C₂×D₄×C₆₇	C₂⁴	1	8	yes	—
100	C₂²×C₄×C₆₇	—	2	2	no	—
101	C₂²×C₄×C₆₇	—	2	2	no	—
102	C₂²×C₄×C₆₇	—	2	16	no	—
103	C₂²×C₄×C₆₇	—	8	8	no	—
104	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
105	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
106	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
108	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
109	Q₁₆×C₆₇	—	2	8	no	—
110	SD₁₆×C₆₇	—	4	16	no	—
111	D₈×C₆₇	—	2	16	no	—
112	D₈×C₆₇	—	4	8	no	—
113	C₂×C₈×C₆₇	—	2	2	no	—
114	C₂×C₈×C₆₇	—	2	4	no	—
115	C₂×C₈×C₆₇	#41	1	4	yes	—
116	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
117	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
118	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
119	C₄⋊C₄×C₆₇	—	2	8	no	—
120	C₄⋊C₄×C₆₇	—	2	2	no	—
121	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
122	C₄⋊C₄×C₆₇	C₂×D₄	1	8	yes	—
123	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
124	C₂²⋊C₄×C₆₇	—	2	16	no	—
125	C₂²⋊C₄×C₆₇	—	2	16	no	—
126	C₂²⋊C₄×C₆₇	—	2	16	no	—
127	C₄²×C₆₇	—	2	8	no	—
128	C₄²×C₆₇	C₂×D₄	1	4	yes	—
129	C₄²×C₆₇	C₂×D₄	1	2	yes	—
130	C₄²×C₆₇	C₂⁴	1	4	yes	—
131	C₄²×C₆₇	C₂⁴	1	2	yes	—
132	C₂³×C₆₇	—	2	8	no	—
133	C₂³×C₆₇	—	4	8	no	—
134	C₂³×C₆₇	—	4	8	no	—
135	C₂³×C₆₇	—	4	8	no	—
136	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
137	Q₈×C₆₇	—	2	8	no	—
138	Q₈×C₆₇	—	2	4	no	—
139	Q₈×C₆₇	—	2	4	no	—
140	Q₈×C₆₇	—	4	4	no	—
141	D₄×C₆₇	—	2	8	no	—
142	D₄×C₆₇	—	2	8	no	—
143	D₄×C₆₇	—	4	8	no	—
144	D₄×C₆₇	—	4	16	no	—
145	D₄×C₆₇	—	8	8	no	—
146	C₂×C₄×C₆₇	—	2	4	no	—
147	C₂×C₄×C₆₇	—	2	4	no	—
148	C₂×C₄×C₆₇	—	2	2	no	—
149	C₂×C₄×C₆₇	—	2	4	no	—
150	C₂×C₄×C₆₇	—	2	4	no	—
151	C₂×C₄×C₆₇	—	4	4	no	—
152	C₂×C₄×C₆₇	—	4	4	no	—
153	C₂×C₄×C₆₇	—	8	16	no	—
154	C₂×C₄×C₆₇	#135	1	4	yes	—
155	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
156	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
157	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
158	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
159	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
160	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
161	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
162	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
163	C₈×C₆₇	—	2	2	no	—
164	C₈×C₆₇	—	2	4	no	—
165	C₈×C₆₇	C₂×#41	1	4	yes	—
166	C₂²×C₆₇	—	4	8	no	—
167	C₂²×C₆₇	—	4	8	no	—
168	C₂²×C₆₇	—	4	8	no	—
169	C₂²×C₆₇	—	8	16	no	—
170	C₂²×C₆₇	#550	1	2	yes	—
171	C₂²×C₆₇	C₂×#135	1	2	yes	—
172	C₂²×C₆₇	#576	1	2	yes	—
173	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
174	C₄×C₆₇	—	2	4	no	—
175	C₄×C₆₇	—	2	4	no	—
176	C₄×C₆₇	—	2	2	no	—
177	C₄×C₆₇	—	4	4	no	—
178	C₄×C₆₇	—	4	4	no	—
179	C₄×C₆₇	#550	1	2	yes	—
180	C₄×C₆₇	C₂×#135	1	4	yes	—
181	C₄×C₆₇	C₂²×#41	1	1	yes	characteristic
182	C₄×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
183	C₂×#3396	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
184	C₂×C₆₇	—	8	8	no	—
185	C₂×C₆₇	#3396	1	4	yes	direct product factor
186	C₂×C₆₇	C₂×#550	1	1	yes	characteristic
187	C₂×C₆₇	C₂²×#135	1	1	yes	characteristic
188	C₂×C₆₇	C₂×#576	1	1	yes	characteristic
189	C₂×#559	C₂×C₆₇	1	2	yes	complemented
190	#3396	C₂×C₆₇	1	16	yes	direct product factor
191	C₂×#495	C₂×C₆₇	1	2	yes	complemented
192	C₂×#493	C₂×C₆₇	1	1	yes	characteristic
193	C₂×#460	C₂×C₆₇	1	2	yes	complemented
194	C₂×#463	C₂×C₆₇	1	4	yes	complemented
195	C₂×C₈×Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
196	C₂×C₄×Q₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
197	C₂×C₄×D₈	C₂×C₆₇	1	2	yes	complemented
198	C₆₇	C₂×#3396	1	1	yes	characteristic, direct product factor, Sylow
199	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
200	C₂²×SD₁₆	C₂²×C₆₇	1	4	yes	—
201	C₂²×D₈	C₂²×C₆₇	1	2	yes	—
202	C₂²×D₈	C₂²×C₆₇	1	2	yes	—
203	#559	—	2	8	no	—
204	#559	C₂²×C₆₇	1	16	yes	complemented
205	C₂×#127	C₂²×C₆₇	1	2	yes	—
206	C₂×#124	—	2	2	no	—
207	C₂×#124	C₂²×C₆₇	1	4	yes	—
208	C₂×#123	—	2	4	no	—
209	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
210	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
211	C₂×C₄×D₄	—	2	2	no	—
212	C₂×C₄×D₄	C₂²×C₆₇	1	4	yes	—
213	#495	C₂²×C₆₇	1	16	yes	complemented
214	#493	C₂²×C₆₇	1	8	yes	—
215	#463	C₂²×C₆₇	1	32	yes	complemented
216	#460	C₂²×C₆₇	1	16	yes	complemented
217	C₈×Q₈	C₂²×C₆₇	1	8	yes	complemented
218	C₄×Q₁₆	C₂²×C₆₇	1	8	yes	complemented
219	C₄×D₈	C₂²×C₆₇	1	16	yes	complemented
220	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
221	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
222	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
223	C₂×#103	C₂²×C₆₇	1	2	yes	—
224	C₂×#102	C₂²×C₆₇	1	4	yes	—
225	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
226	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
227	C₂²×Q₈	D₄×C₆₇	1	1	yes	characteristic
228	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
229	C₂²×D₄	—	2	2	no	—
230	C₂²×D₄	—	2	2	no	—
231	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
232	C₂³×C₄	—	2	4	no	—
233	C₂×Q₁₆	—	2	4	no	—
234	C₂×Q₁₆	C₂³×C₆₇	1	4	yes	—
235	C₂×SD₁₆	—	4	8	no	—
236	C₂×SD₁₆	C₂³×C₆₇	1	16	yes	—
237	C₂×D₈	—	2	8	no	—
238	C₂×D₈	—	4	4	no	—
239	C₂×D₈	C₂³×C₆₇	1	8	yes	—
240	C₂×D₈	C₂³×C₆₇	1	8	yes	—
241	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
242	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
243	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
244	#127	—	2	4	no	—
245	#127	C₂³×C₆₇	1	8	yes	—
246	#124	—	2	16	no	—
247	#124	—	2	8	no	—
248	#124	C₂³×C₆₇	1	16	yes	—
249	#123	—	2	32	no	—
250	C₄×Q₈	—	2	4	no	—
251	C₄×Q₈	D₄×C₆₇	1	4	yes	—
252	C₄×Q₈	C₂³×C₆₇	1	8	yes	—
253	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
254	C₄×D₄	—	2	16	no	—
255	C₄×D₄	—	2	8	no	—
256	C₄×D₄	C₂³×C₆₇	1	16	yes	—
257	C₂×C₄⋊C₄	—	2	2	no	—
258	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
259	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
260	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
261	C₂×C₂²⋊C₄	—	2	4	no	—
262	C₂×C₂²⋊C₄	—	2	4	no	—
263	C₂×C₂²⋊C₄	—	2	4	no	—
264	C₂×C₄²	—	2	2	no	—
265	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
266	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
267	#107	C₂³×C₆₇	1	4	yes	—
268	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
269	C₄⋊C₈	C₂³×C₆₇	1	8	yes	—
270	#103	C₂³×C₆₇	1	8	yes	—
271	#102	C₂³×C₆₇	1	16	yes	—
272	C₄×C₈	C₂³×C₆₇	1	8	yes	—
273	C₄×C₈	C₂³×C₆₇	1	4	yes	—
274	C₂⁴	—	2	4	no	—
275	C₂×Q₈	—	2	4	no	—
276	C₂×Q₈	—	2	2	no	—
277	C₂×Q₈	—	2	4	no	—
278	C₂×Q₈	—	2	4	no	—
279	C₂×Q₈	C₂×D₄×C₆₇	1	4	yes	—
280	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
281	C₂×D₄	—	2	4	no	—
282	C₂×D₄	—	2	8	no	—
283	C₂×D₄	—	2	8	no	—
284	C₂×D₄	—	2	8	no	—
285	C₂×D₄	—	2	8	no	—
286	C₂×D₄	—	4	8	no	—
287	C₂×D₄	—	8	4	no	—
288	C₂×D₄	C₂⁴×C₆₇	1	8	yes	—
289	C₂²×C₄	—	2	2	no	—
290	C₂²×C₄	—	2	2	no	—
291	C₂²×C₄	—	2	16	no	—
292	C₂²×C₄	—	8	8	no	—
293	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
294	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
295	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
296	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
297	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
298	Q₁₆	—	2	8	no	—
299	SD₁₆	—	4	16	no	—
300	D₈	—	2	16	no	—
301	D₈	—	4	8	no	—
302	C₂×C₈	—	2	2	no	—
303	C₂×C₈	—	2	4	no	—
304	C₂×C₈	#41×C₆₇	1	4	yes	—
305	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
306	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
307	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
308	C₄⋊C₄	—	2	8	no	—
309	C₄⋊C₄	—	2	2	no	—
310	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
311	C₄⋊C₄	C₂×D₄×C₆₇	1	8	yes	—
312	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
313	C₂²⋊C₄	—	2	16	no	—
314	C₂²⋊C₄	—	2	16	no	—
315	C₂²⋊C₄	—	2	16	no	—
316	C₄²	—	2	8	no	—
317	C₄²	C₂×D₄×C₆₇	1	4	yes	—
318	C₄²	C₂×D₄×C₆₇	1	2	yes	—
319	C₄²	C₂⁴×C₆₇	1	4	yes	—
320	C₄²	C₂⁴×C₆₇	1	2	yes	—
321	C₂³	—	2	8	no	—
322	C₂³	—	4	8	no	—
323	C₂³	—	4	8	no	—
324	C₂³	—	4	8	no	—
325	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
326	Q₈	—	2	8	no	—
327	Q₈	—	2	4	no	—
328	Q₈	—	2	4	no	—
329	Q₈	—	4	4	no	—
330	D₄	—	2	8	no	—
331	D₄	—	2	8	no	—
332	D₄	—	4	8	no	—
333	D₄	—	4	16	no	—
334	D₄	—	8	8	no	—
335	C₂×C₄	—	2	4	no	—
336	C₂×C₄	—	2	4	no	—
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	2	4	no	—
339	C₂×C₄	—	2	4	no	—
340	C₂×C₄	—	4	4	no	—
341	C₂×C₄	—	4	4	no	—
342	C₂×C₄	—	8	16	no	—
343	C₂×C₄	#135×C₆₇	1	4	yes	—
344	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
345	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
346	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
347	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
348	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
349	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
350	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
351	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
352	C₈	—	2	2	no	—
353	C₈	—	2	4	no	—
354	C₈	C₂×#41×C₆₇	1	4	yes	—
355	C₂²	—	4	8	no	—
356	C₂²	—	4	8	no	—
357	C₂²	—	4	8	no	—
358	C₂²	—	8	16	no	—
359	C₂²	#550×C₆₇	1	2	yes	—
360	C₂²	C₂×#135×C₆₇	1	2	yes	—
361	C₂²	#576×C₆₇	1	2	yes	—
362	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
363	C₄	—	2	4	no	—
364	C₄	—	2	4	no	—
365	C₄	—	2	2	no	—
366	C₄	—	4	4	no	—
367	C₄	—	4	4	no	—
368	C₄	#550×C₆₇	1	2	yes	—
369	C₄	C₂×#135×C₆₇	1	4	yes	—
370	C₄	C₂²×#41×C₆₇	1	1	yes	characteristic
371	C₄	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
372	C₂	—	8	8	no	—
373	C₂	C₆₇×#3396	1	4	yes	direct product factor
374	C₂	C₂×#550×C₆₇	1	1	yes	characteristic
375	C₂	C₂²×#135×C₆₇	1	1	yes	characteristic
376	C₂	C₂×#576×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
prime radical	C₂×#493×C₆₇	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×C₄×Q₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#559×C₆₇	C₂	semidirect
2. C₆₇×#3396	C₂	direct (N × H)
3. C₂×#495×C₆₇	C₂	semidirect
5. C₂×#460×C₆₇	C₂	semidirect
6. C₂×#463×C₆₇	C₂	semidirect
7. C₂×C₈×Q₈×C₆₇	C₂	semidirect
8. C₂×C₄×Q₁₆×C₆₇	C₂	semidirect
9. C₂×C₄×D₈×C₆₇	C₂	semidirect
15. #559×C₆₇	C₂²	semidirect
24. #495×C₆₇	C₂²	semidirect
26. #463×C₆₇	C₂²	semidirect
27. #460×C₆₇	C₂²	semidirect
28. C₈×Q₈×C₆₇	C₂²	semidirect
29. C₄×Q₁₆×C₆₇	C₂²	semidirect
30. C₄×D₈×C₆₇	C₂²	semidirect
183. C₂×#3396	C₆₇	direct (N × H)
185. C₂×C₆₇	#3396	direct (N × H)
189. C₂×#559	C₂×C₆₇	semidirect
190. #3396	C₂×C₆₇	direct (N × H)
191. C₂×#495	C₂×C₆₇	semidirect
193. C₂×#460	C₂×C₆₇	semidirect
194. C₂×#463	C₂×C₆₇	semidirect
195. C₂×C₈×Q₈	C₂×C₆₇	semidirect
196. C₂×C₄×Q₁₆	C₂×C₆₇	semidirect
197. C₂×C₄×D₈	C₂×C₆₇	semidirect
198. C₆₇	C₂×#3396	direct (N × H)
204. #559	C₂²×C₆₇	semidirect
213. #495	C₂²×C₆₇	semidirect
215. #463	C₂²×C₆₇	semidirect
216. #460	C₂²×C₆₇	semidirect
217. C₈×Q₈	C₂²×C₆₇	semidirect
218. C₄×Q₁₆	C₂²×C₆₇	semidirect
219. C₄×D₈	C₂²×C₆₇	semidirect
373. C₂	C₆₇×#3396	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×#559×C₆₇	2
C₆₇×#3396	2
C₂×#495×C₆₇	2
C₂×#493×C₆₇	2
C₂×#460×C₆₇	2
C₂×#463×C₆₇	2
C₂×C₈×Q₈×C₆₇	2
C₂×C₄×Q₁₆×C₆₇	2
C₂×C₄×D₈×C₆₇	2
C₂×#3396	67

Subgroups — C2×C67×#3396 #30740643

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#3396 #30740643