Subgroups — C₂×C₆₇×#3283 #30740294

Order 17152.

2806 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#135×C₆₇	C₂	1	2	yes	maximal, complemented
2	C₂×#547×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₆₇×#3283	C₂	1	16	yes	maximal, direct product factor
4	C₂×#494×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#467×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₂×#461×C₆₇	C₂	1	2	yes	maximal, complemented
7	C₂×#452×C₆₇	C₂	1	4	yes	maximal, complemented
8	C₂×C₄×D₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₈×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#41×C₆₇	C₂²	1	4	yes	—
11	C₂×#135×C₆₇	—	2	16	no	—
12	C₂×#135×C₆₇	C₂²	1	16	yes	complemented
13	C₂²×Q₁₆×C₆₇	C₂²	1	2	yes	—
14	C₂²×Q₁₆×C₆₇	C₂²	1	2	yes	—
15	C₂²×SD₁₆×C₆₇	C₂²	1	4	yes	—
16	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
18	#547×C₆₇	—	2	8	no	—
19	#547×C₆₇	C₂²	1	16	yes	complemented
20	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
21	C₂×#123×C₆₇	—	2	4	no	—
22	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	4	yes	—
23	C₂×#121×C₆₇	—	2	2	no	—
24	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
25	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
26	C₂²×C₄⋊C₄×C₆₇	—	2	2	no	—
27	#494×C₆₇	C₂²	1	8	yes	complemented
28	#467×C₆₇	C₂²	1	16	yes	complemented
29	#461×C₆₇	C₂²	1	16	yes	complemented
30	#452×C₆₇	C₂²	1	32	yes	complemented
31	C₄×D₈×C₆₇	C₂²	1	8	yes	complemented
32	C₈×D₄×C₆₇	C₂²	1	8	yes	complemented
33	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
34	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
35	C₂×#103×C₆₇	C₂²	1	4	yes	—
36	C₂×#102×C₆₇	C₂²	1	2	yes	—
37	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
38	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
39	C₂×#41×C₆₇	—	2	8	no	—
40	C₂×#41×C₆₇	—	2	16	no	—
41	C₂×#41×C₆₇	—	4	8	no	—
42	C₂×#41×C₆₇	C₂³	1	16	yes	—
43	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
44	C₂²×D₄×C₆₇	—	2	4	no	—
45	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
46	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
47	C₂³×C₄×C₆₇	—	2	2	no	—
48	C₂³×C₄×C₆₇	—	2	4	no	—
49	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
50	#135×C₆₇	—	2	64	no	—
51	C₂×Q₁₆×C₆₇	—	2	8	no	—
52	C₂×Q₁₆×C₆₇	—	4	4	no	—
53	C₂×Q₁₆×C₆₇	C₂³	1	8	yes	—
54	C₂×Q₁₆×C₆₇	C₂³	1	8	yes	—
55	C₂×SD₁₆×C₆₇	—	2	16	no	—
56	C₂×SD₁₆×C₆₇	C₂³	1	16	yes	—
57	C₂×D₈×C₆₇	D₄	1	8	yes	complemented
58	C₂×D₈×C₆₇	C₂³	1	4	yes	—
59	C₂²×C₈×C₆₇	—	2	8	no	—
60	C₂²×C₈×C₆₇	C₂³	1	8	yes	—
61	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
62	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
63	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
64	C₄⋊Q₈×C₆₇	—	2	4	no	—
65	C₄⋊Q₈×C₆₇	C₂³	1	8	yes	—
66	#123×C₆₇	—	2	32	no	—
67	C₂²⋊Q₈×C₆₇	—	2	8	no	—
68	C₂²⋊Q₈×C₆₇	C₂³	1	16	yes	—
69	#121×C₆₇	—	2	16	no	—
70	C₄×D₄×C₆₇	—	2	4	no	—
71	C₄×D₄×C₆₇	D₄	1	4	yes	—
72	C₄×D₄×C₆₇	C₂³	1	8	yes	—
73	C₄×D₄×C₆₇	C₂³	1	4	yes	—
74	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
75	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
76	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
77	C₂×C₄⋊C₄×C₆₇	—	2	16	no	—
78	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
79	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
80	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
81	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
82	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
83	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
84	#107×C₆₇	C₂³	1	4	yes	—
85	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
86	#103×C₆₇	C₂³	1	16	yes	—
87	#102×C₆₇	C₂³	1	8	yes	—
88	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	—
89	C₄×C₈×C₆₇	C₂³	1	4	yes	—
90	C₂⁴×C₆₇	—	2	2	no	—
91	C₂⁴×C₆₇	C₂×D₄	1	2	yes	—
92	#41×C₆₇	—	2	32	no	—
93	#41×C₆₇	—	4	16	no	—
94	#41×C₆₇	—	4	32	no	—
95	C₂×Q₈×C₆₇	—	2	4	no	—
96	C₂×Q₈×C₆₇	—	2	8	no	—
97	C₂×Q₈×C₆₇	—	2	8	no	—
98	C₂×Q₈×C₆₇	C₂⁴	1	8	yes	—
99	C₂×D₄×C₆₇	—	2	16	no	—
100	C₂×D₄×C₆₇	—	2	4	no	—
101	C₂×D₄×C₆₇	—	2	2	no	—
102	C₂×D₄×C₆₇	—	2	4	no	—
103	C₂×D₄×C₆₇	—	4	16	no	—
104	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
105	C₂×D₄×C₆₇	C₂×D₄	1	8	yes	—
106	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
107	C₂²×C₄×C₆₇	—	2	4	no	—
108	C₂²×C₄×C₆₇	—	2	4	no	—
109	C₂²×C₄×C₆₇	—	2	2	no	—
110	C₂²×C₄×C₆₇	—	2	4	no	—
111	C₂²×C₄×C₆₇	—	2	8	no	—
112	C₂²×C₄×C₆₇	—	2	16	no	—
113	C₂²×C₄×C₆₇	—	4	16	no	—
114	C₂²×C₄×C₆₇	—	4	8	no	—
115	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
116	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
118	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
120	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
121	Q₁₆×C₆₇	—	2	16	no	—
122	Q₁₆×C₆₇	—	4	8	no	—
123	SD₁₆×C₆₇	—	2	32	no	—
124	D₈×C₆₇	C₂×D₄	1	16	yes	complemented
125	C₂×C₈×C₆₇	—	2	4	no	—
126	C₂×C₈×C₆₇	—	2	2	no	—
127	C₂×C₈×C₆₇	—	2	16	no	—
128	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
129	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
130	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
131	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
132	C₄⋊C₄×C₆₇	—	2	8	no	—
133	C₄⋊C₄×C₆₇	—	2	16	no	—
134	C₄⋊C₄×C₆₇	—	2	8	no	—
135	C₄⋊C₄×C₆₇	—	2	2	no	—
136	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
137	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
138	C₂²⋊C₄×C₆₇	—	2	16	no	—
139	C₂²⋊C₄×C₆₇	—	2	8	no	—
140	C₂²⋊C₄×C₆₇	C₂×D₄	1	8	yes	—
141	C₄²×C₆₇	C₂×D₄	1	2	yes	—
142	C₄²×C₆₇	C₂⁴	1	2	yes	—
143	C₂³×C₆₇	—	2	4	no	—
144	C₂³×C₆₇	—	2	8	no	—
145	C₂³×C₆₇	—	2	4	no	—
146	C₂³×C₆₇	—	2	4	no	—
147	C₂³×C₆₇	—	2	4	no	—
148	C₂³×C₆₇	—	4	4	no	—
149	C₂³×C₆₇	—	4	4	no	—
150	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
151	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
152	Q₈×C₆₇	—	2	8	no	—
153	Q₈×C₆₇	—	2	8	no	—
154	Q₈×C₆₇	—	4	8	no	—
155	D₄×C₆₇	—	2	8	no	—
156	D₄×C₆₇	—	2	8	no	—
157	D₄×C₆₇	—	2	4	no	—
158	D₄×C₆₇	—	4	32	no	—
159	D₄×C₆₇	C₂²×D₄	1	8	yes	—
160	C₂×C₄×C₆₇	—	2	4	no	—
161	C₂×C₄×C₆₇	—	2	2	no	—
162	C₂×C₄×C₆₇	—	2	8	no	—
163	C₂×C₄×C₆₇	—	2	4	no	—
164	C₂×C₄×C₆₇	—	2	4	no	—
165	C₂×C₄×C₆₇	—	2	4	no	—
166	C₂×C₄×C₆₇	—	2	8	no	—
167	C₂×C₄×C₆₇	—	4	32	no	—
168	C₂×C₄×C₆₇	—	4	8	no	—
169	C₂×C₄×C₆₇	—	4	4	no	—
170	C₂×C₄×C₆₇	—	4	16	no	—
171	C₂×C₄×C₆₇	#135	1	4	yes	complemented
172	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
173	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
174	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
175	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
176	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
177	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
178	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
179	C₈×C₆₇	—	2	4	no	—
180	C₈×C₆₇	—	2	2	no	—
181	C₈×C₆₇	C₂²×D₄	1	4	yes	—
182	C₂²×C₆₇	—	2	4	no	—
183	C₂²×C₆₇	—	2	8	no	—
184	C₂²×C₆₇	—	2	4	no	—
185	C₂²×C₆₇	—	2	4	no	—
186	C₂²×C₆₇	—	4	8	no	—
187	C₂²×C₆₇	—	4	16	no	—
188	C₂²×C₆₇	—	4	4	no	—
189	C₂²×C₆₇	—	4	4	no	—
190	C₂²×C₆₇	D₄²	1	2	yes	—
191	C₂²×C₆₇	C₂×#135	1	2	yes	—
192	C₂²×C₆₇	#578	1	2	yes	—
193	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
194	C₄×C₆₇	—	2	4	no	—
195	C₄×C₆₇	—	2	2	no	—
196	C₄×C₆₇	—	4	8	no	—
197	C₄×C₆₇	—	4	4	no	—
198	C₄×C₆₇	D₄²	1	2	yes	—
199	C₄×C₆₇	C₂×#135	1	4	yes	complemented
200	C₄×C₆₇	C₂³×D₄	1	1	yes	characteristic
201	C₄×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
202	C₂×#3283	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
203	C₂×C₆₇	—	4	4	no	—
204	C₂×C₆₇	—	4	8	no	—
205	C₂×C₆₇	#3283	1	4	yes	direct product factor
206	C₂×C₆₇	C₂×D₄²	1	1	yes	characteristic
207	C₂×C₆₇	C₂²×#135	1	1	yes	characteristic
208	C₂×C₆₇	C₂×#578	1	1	yes	characteristic
209	C₂²×#135	C₂×C₆₇	1	2	yes	complemented
210	C₂×#547	C₂×C₆₇	1	2	yes	complemented
211	#3283	C₂×C₆₇	1	16	yes	direct product factor
212	C₂×#494	C₂×C₆₇	1	1	yes	characteristic, complemented
213	C₂×#467	C₂×C₆₇	1	2	yes	complemented
214	C₂×#461	C₂×C₆₇	1	2	yes	complemented
215	C₂×#452	C₂×C₆₇	1	4	yes	complemented
216	C₂×C₄×D₈	C₂×C₆₇	1	1	yes	characteristic, complemented
217	C₂×C₈×D₄	C₂×C₆₇	1	1	yes	characteristic, complemented
218	C₆₇	C₂×#3283	1	1	yes	characteristic, direct product factor, Sylow
219	C₂²×#41	C₂²×C₆₇	1	4	yes	—
220	C₂×#135	—	2	16	no	—
221	C₂×#135	C₂²×C₆₇	1	16	yes	complemented
222	C₂²×Q₁₆	C₂²×C₆₇	1	2	yes	—
223	C₂²×Q₁₆	C₂²×C₆₇	1	2	yes	—
224	C₂²×SD₁₆	C₂²×C₆₇	1	4	yes	—
225	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
226	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
227	#547	—	2	8	no	—
228	#547	C₂²×C₆₇	1	16	yes	complemented
229	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
230	C₂×#123	—	2	4	no	—
231	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	4	yes	—
232	C₂×#121	—	2	2	no	—
233	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
234	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
235	C₂²×C₄⋊C₄	—	2	2	no	—
236	#494	C₂²×C₆₇	1	8	yes	complemented
237	#467	C₂²×C₆₇	1	16	yes	complemented
238	#461	C₂²×C₆₇	1	16	yes	complemented
239	#452	C₂²×C₆₇	1	32	yes	complemented
240	C₄×D₈	C₂²×C₆₇	1	8	yes	complemented
241	C₈×D₄	C₂²×C₆₇	1	8	yes	complemented
242	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
243	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
244	C₂×#103	C₂²×C₆₇	1	4	yes	—
245	C₂×#102	C₂²×C₆₇	1	2	yes	—
246	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
247	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
248	C₂×#41	—	2	8	no	—
249	C₂×#41	—	2	16	no	—
250	C₂×#41	—	4	8	no	—
251	C₂×#41	C₂³×C₆₇	1	16	yes	—
252	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
253	C₂²×D₄	—	2	4	no	—
254	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
255	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
256	C₂³×C₄	—	2	2	no	—
257	C₂³×C₄	—	2	4	no	—
258	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
259	#135	—	2	64	no	—
260	C₂×Q₁₆	—	2	8	no	—
261	C₂×Q₁₆	—	4	4	no	—
262	C₂×Q₁₆	C₂³×C₆₇	1	8	yes	—
263	C₂×Q₁₆	C₂³×C₆₇	1	8	yes	—
264	C₂×SD₁₆	—	2	16	no	—
265	C₂×SD₁₆	C₂³×C₆₇	1	16	yes	—
266	C₂×D₈	D₄×C₆₇	1	8	yes	complemented
267	C₂×D₈	C₂³×C₆₇	1	4	yes	—
268	C₂²×C₈	—	2	8	no	—
269	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
270	C₂²×C₈	C₂³×C₆₇	1	8	yes	—
271	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
272	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
273	C₄⋊Q₈	—	2	4	no	—
274	C₄⋊Q₈	C₂³×C₆₇	1	8	yes	—
275	#123	—	2	32	no	—
276	C₂²⋊Q₈	—	2	8	no	—
277	C₂²⋊Q₈	C₂³×C₆₇	1	16	yes	—
278	#121	—	2	16	no	—
279	C₄×D₄	—	2	4	no	—
280	C₄×D₄	D₄×C₆₇	1	4	yes	—
281	C₄×D₄	C₂³×C₆₇	1	8	yes	—
282	C₄×D₄	C₂³×C₆₇	1	4	yes	—
283	C₂×C₄⋊C₄	—	2	2	no	—
284	C₂×C₄⋊C₄	—	2	4	no	—
285	C₂×C₄⋊C₄	—	2	2	no	—
286	C₂×C₄⋊C₄	—	2	16	no	—
287	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
288	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
289	C₂×C₂²⋊C₄	—	2	4	no	—
290	C₂×C₂²⋊C₄	—	2	2	no	—
291	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
292	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
293	#107	C₂³×C₆₇	1	4	yes	—
294	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
295	#103	C₂³×C₆₇	1	16	yes	—
296	#102	C₂³×C₆₇	1	8	yes	—
297	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	—
298	C₄×C₈	C₂³×C₆₇	1	4	yes	—
299	C₂⁴	—	2	2	no	—
300	C₂⁴	C₂×D₄×C₆₇	1	2	yes	—
301	#41	—	2	32	no	—
302	#41	—	4	16	no	—
303	#41	—	4	32	no	—
304	C₂×Q₈	—	2	4	no	—
305	C₂×Q₈	—	2	8	no	—
306	C₂×Q₈	—	2	8	no	—
307	C₂×Q₈	C₂⁴×C₆₇	1	8	yes	—
308	C₂×D₄	—	2	16	no	—
309	C₂×D₄	—	2	4	no	—
310	C₂×D₄	—	2	2	no	—
311	C₂×D₄	—	2	4	no	—
312	C₂×D₄	—	4	16	no	—
313	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
314	C₂×D₄	C₂×D₄×C₆₇	1	8	yes	—
315	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
316	C₂²×C₄	—	2	4	no	—
317	C₂²×C₄	—	2	4	no	—
318	C₂²×C₄	—	2	2	no	—
319	C₂²×C₄	—	2	4	no	—
320	C₂²×C₄	—	2	8	no	—
321	C₂²×C₄	—	2	16	no	—
322	C₂²×C₄	—	4	16	no	—
323	C₂²×C₄	—	4	8	no	—
324	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
325	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
326	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
327	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
328	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
329	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
330	Q₁₆	—	2	16	no	—
331	Q₁₆	—	4	8	no	—
332	SD₁₆	—	2	32	no	—
333	D₈	C₂×D₄×C₆₇	1	16	yes	complemented
334	C₂×C₈	—	2	4	no	—
335	C₂×C₈	—	2	2	no	—
336	C₂×C₈	—	2	16	no	—
337	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
338	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
339	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
340	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
341	C₄⋊C₄	—	2	8	no	—
342	C₄⋊C₄	—	2	16	no	—
343	C₄⋊C₄	—	2	8	no	—
344	C₄⋊C₄	—	2	2	no	—
345	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
346	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
347	C₂²⋊C₄	—	2	16	no	—
348	C₂²⋊C₄	—	2	8	no	—
349	C₂²⋊C₄	C₂×D₄×C₆₇	1	8	yes	—
350	C₄²	C₂×D₄×C₆₇	1	2	yes	—
351	C₄²	C₂⁴×C₆₇	1	2	yes	—
352	C₂³	—	2	4	no	—
353	C₂³	—	2	8	no	—
354	C₂³	—	2	4	no	—
355	C₂³	—	2	4	no	—
356	C₂³	—	2	4	no	—
357	C₂³	—	4	4	no	—
358	C₂³	—	4	4	no	—
359	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
360	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
361	Q₈	—	2	8	no	—
362	Q₈	—	2	8	no	—
363	Q₈	—	4	8	no	—
364	D₄	—	2	8	no	—
365	D₄	—	2	8	no	—
366	D₄	—	2	4	no	—
367	D₄	—	4	32	no	—
368	D₄	C₂²×D₄×C₆₇	1	8	yes	—
369	C₂×C₄	—	2	4	no	—
370	C₂×C₄	—	2	2	no	—
371	C₂×C₄	—	2	8	no	—
372	C₂×C₄	—	2	4	no	—
373	C₂×C₄	—	2	4	no	—
374	C₂×C₄	—	2	4	no	—
375	C₂×C₄	—	2	8	no	—
376	C₂×C₄	—	4	32	no	—
377	C₂×C₄	—	4	8	no	—
378	C₂×C₄	—	4	4	no	—
379	C₂×C₄	—	4	16	no	—
380	C₂×C₄	#135×C₆₇	1	4	yes	complemented
381	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
382	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
383	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
384	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
385	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
386	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
387	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
388	C₈	—	2	4	no	—
389	C₈	—	2	2	no	—
390	C₈	C₂²×D₄×C₆₇	1	4	yes	—
391	C₂²	—	2	4	no	—
392	C₂²	—	2	4	no	—
393	C₂²	—	2	4	no	—
394	C₂²	—	2	8	no	—
395	C₂²	—	4	8	no	—
396	C₂²	—	4	16	no	—
397	C₂²	—	4	4	no	—
398	C₂²	—	4	4	no	—
399	C₂²	D₄²×C₆₇	1	2	yes	—
400	C₂²	C₂×#135×C₆₇	1	2	yes	—
401	C₂²	#578×C₆₇	1	2	yes	—
402	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
403	C₄	—	2	2	no	—
404	C₄	—	2	4	no	—
405	C₄	—	4	4	no	—
406	C₄	—	4	8	no	—
407	C₄	D₄²×C₆₇	1	2	yes	—
408	C₄	C₂×#135×C₆₇	1	4	yes	complemented
409	C₄	C₂³×D₄×C₆₇	1	1	yes	characteristic
410	C₄	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
411	C₂	—	4	4	no	—
412	C₂	—	4	8	no	—
413	C₂	C₆₇×#3283	1	4	yes	direct product factor
414	C₂	C₂×D₄²×C₆₇	1	1	yes	characteristic
415	C₂	C₂²×#135×C₆₇	1	1	yes	characteristic
416	C₂	C₂×#578×C₆₇	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂²×#135×C₆₇	C₂	semidirect
2. C₂×#547×C₆₇	C₂	semidirect
3. C₆₇×#3283	C₂	direct (N × H)
4. C₂×#494×C₆₇	C₂	semidirect
5. C₂×#467×C₆₇	C₂	semidirect
6. C₂×#461×C₆₇	C₂	semidirect
7. C₂×#452×C₆₇	C₂	semidirect
8. C₂×C₄×D₈×C₆₇	C₂	semidirect
9. C₂×C₈×D₄×C₆₇	C₂	semidirect
12. C₂×#135×C₆₇	C₂²	semidirect
19. #547×C₆₇	C₂²	semidirect
27. #494×C₆₇	C₂²	semidirect
28. #467×C₆₇	C₂²	semidirect
29. #461×C₆₇	C₂²	semidirect
30. #452×C₆₇	C₂²	semidirect
31. C₄×D₈×C₆₇	C₂²	semidirect
32. C₈×D₄×C₆₇	C₂²	semidirect
57. C₂×D₈×C₆₇	D₄	semidirect
124. D₈×C₆₇	C₂×D₄	semidirect
171. C₂×C₄×C₆₇	#135	semidirect
199. C₄×C₆₇	C₂×#135	semidirect
202. C₂×#3283	C₆₇	direct (N × H)
205. C₂×C₆₇	#3283	direct (N × H)
209. C₂²×#135	C₂×C₆₇	semidirect
210. C₂×#547	C₂×C₆₇	semidirect
211. #3283	C₂×C₆₇	direct (N × H)
212. C₂×#494	C₂×C₆₇	semidirect
213. C₂×#467	C₂×C₆₇	semidirect
214. C₂×#461	C₂×C₆₇	semidirect
215. C₂×#452	C₂×C₆₇	semidirect
216. C₂×C₄×D₈	C₂×C₆₇	semidirect
217. C₂×C₈×D₄	C₂×C₆₇	semidirect
218. C₆₇	C₂×#3283	direct (N × H)
221. C₂×#135	C₂²×C₆₇	semidirect
228. #547	C₂²×C₆₇	semidirect
236. #494	C₂²×C₆₇	semidirect
237. #467	C₂²×C₆₇	semidirect
238. #461	C₂²×C₆₇	semidirect
239. #452	C₂²×C₆₇	semidirect
240. C₄×D₈	C₂²×C₆₇	semidirect
241. C₈×D₄	C₂²×C₆₇	semidirect
266. C₂×D₈	D₄×C₆₇	semidirect
333. D₈	C₂×D₄×C₆₇	semidirect
380. C₂×C₄	#135×C₆₇	semidirect
408. C₄	C₂×#135×C₆₇	semidirect
413. C₂	C₆₇×#3283	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×#135×C₆₇	2
C₂×#547×C₆₇	2
C₆₇×#3283	2
C₂×#494×C₆₇	2
C₂×#467×C₆₇	2
C₂×#461×C₆₇	2
C₂×#452×C₆₇	2
C₂×C₄×D₈×C₆₇	2
C₂×C₈×D₄×C₆₇	2
C₂×#3283	67

Subgroups — C2×C67×#3283 #30740294

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#3283 #30740294