Subgroups — C₂×C₆₇×#2820 #30715932

Order 17152.

2566 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#125×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#124×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2820	C₂	1	16	yes	maximal, direct product factor
4	C₂×#399×C₆₇	C₂	1	2	yes	maximal, complemented
5	C₂×#397×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₂×#396×C₆₇	C₂	1	4	yes	maximal, complemented
7	C₂×#395×C₆₇	C₂	1	2	yes	maximal, complemented
8	C₂×#394×C₆₇	C₂	1	2	yes	maximal
9	C₂×#382×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
11	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×#125×C₆₇	—	2	8	no	—
13	C₂×#125×C₆₇	C₂²	1	8	yes	complemented
14	C₂×#124×C₆₇	—	2	8	no	—
15	C₂×#124×C₆₇	C₂²	1	8	yes	—
16	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
17	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
18	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
19	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
20	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
21	C₂²×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
22	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
23	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
24	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
25	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
26	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
27	#399×C₆₇	C₂²	1	16	yes	complemented
28	#397×C₆₇	C₂²	1	16	yes	complemented
29	#396×C₆₇	C₂²	1	32	yes	complemented
30	#395×C₆₇	C₂²	1	16	yes	complemented
31	#394×C₆₇	C₂²	1	16	yes	—
32	#382×C₆₇	C₂²	1	8	yes	complemented
33	C₂×#95×C₆₇	C₂²	1	2	yes	—
34	C₂×#95×C₆₇	C₂²	1	2	yes	—
35	C₂×#95×C₆₇	C₂²	1	4	yes	—
36	C₂⁵×C₆₇	C₂³	1	2	yes	—
37	C₂²×Q₈×C₆₇	—	2	4	no	—
38	C₂²×Q₈×C₆₇	—	4	4	no	—
39	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
40	C₂²×D₄×C₆₇	—	2	4	no	—
41	C₂²×D₄×C₆₇	—	4	4	no	—
42	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
43	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
44	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
45	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
46	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
47	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
48	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
49	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
50	#125×C₆₇	—	2	32	no	—
51	#124×C₆₇	—	2	32	no	—
52	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
53	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
54	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
55	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
56	C₂×C₄⋊C₄×C₆₇	—	4	4	no	—
57	C₂×C₄⋊C₄×C₆₇	—	4	8	no	—
58	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
59	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
60	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
61	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
62	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
63	C₂×C₄⋊C₄×C₆₇	C₂³	1	16	yes	—
64	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
65	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
66	C₂×C₂²⋊C₄×C₆₇	—	4	8	no	—
67	C₂×C₂²⋊C₄×C₆₇	—	4	8	no	—
68	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
69	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
70	C₂×C₂²⋊C₄×C₆₇	C₂³	1	16	yes	—
71	C₂×C₂²⋊C₄×C₆₇	C₂³	1	16	yes	—
72	C₂×C₄²×C₆₇	D₄	1	8	yes	—
73	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
74	#95×C₆₇	C₂³	1	8	yes	—
75	#95×C₆₇	C₂³	1	8	yes	—
76	#95×C₆₇	C₂³	1	16	yes	—
77	C₂⁴×C₆₇	—	2	4	no	—
78	C₂⁴×C₆₇	—	2	4	no	—
79	C₂⁴×C₆₇	—	2	8	no	—
80	C₂⁴×C₆₇	—	2	8	no	—
81	C₂⁴×C₆₇	—	2	4	no	—
82	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
83	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
84	C₂×Q₈×C₆₇	—	2	8	no	—
85	C₂×Q₈×C₆₇	—	4	8	no	—
86	C₂×Q₈×C₆₇	—	4	16	no	—
87	C₂×D₄×C₆₇	—	2	8	no	—
88	C₂×D₄×C₆₇	—	4	8	no	—
89	C₂×D₄×C₆₇	—	4	16	no	—
90	C₂²×C₄×C₆₇	—	2	2	no	—
91	C₂²×C₄×C₆₇	—	2	4	no	—
92	C₂²×C₄×C₆₇	—	2	4	no	—
93	C₂²×C₄×C₆₇	—	2	2	no	—
94	C₂²×C₄×C₆₇	—	2	4	no	—
95	C₂²×C₄×C₆₇	—	2	4	no	—
96	C₂²×C₄×C₆₇	—	2	4	no	—
97	C₂²×C₄×C₆₇	—	2	4	no	—
98	C₂²×C₄×C₆₇	—	2	8	no	—
99	C₂²×C₄×C₆₇	—	2	8	no	—
100	C₂²×C₄×C₆₇	—	2	8	no	—
101	C₂²×C₄×C₆₇	—	2	8	no	—
102	C₂²×C₄×C₆₇	—	2	4	no	—
103	C₂²×C₄×C₆₇	—	2	8	no	—
104	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
105	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
106	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
107	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
108	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
109	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
110	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
111	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
112	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
113	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
114	C₄⋊C₄×C₆₇	—	2	16	no	—
115	C₄⋊C₄×C₆₇	—	2	8	no	—
116	C₄⋊C₄×C₆₇	—	2	8	no	—
117	C₄⋊C₄×C₆₇	—	2	16	no	—
118	C₄⋊C₄×C₆₇	—	4	8	no	—
119	C₄⋊C₄×C₆₇	—	4	16	no	—
120	C₂²⋊C₄×C₆₇	—	2	16	no	—
121	C₂²⋊C₄×C₆₇	—	2	16	no	—
122	C₂²⋊C₄×C₆₇	—	4	16	no	—
123	C₂²⋊C₄×C₆₇	—	4	16	no	—
124	C₄²×C₆₇	C₂×D₄	1	16	yes	—
125	C₂³×C₆₇	—	2	4	no	—
126	C₂³×C₆₇	—	2	4	no	—
127	C₂³×C₆₇	—	2	8	no	—
128	C₂³×C₆₇	—	2	8	no	—
129	C₂³×C₆₇	—	2	4	no	—
130	C₂³×C₆₇	—	4	8	no	—
131	C₂³×C₆₇	—	4	8	no	—
132	C₂³×C₆₇	—	4	8	no	—
133	C₂³×C₆₇	—	4	16	no	—
134	C₂³×C₆₇	—	4	16	no	—
135	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
136	C₂³×C₆₇	H(5,𝔽₂)	1	4	yes	—
137	C₂³×C₆₇	H(5,𝔽₂)	1	2	yes	—
138	C₂³×C₆₇	2₋¹⁺⁴	1	4	yes	—
139	C₂³×C₆₇	2₋¹⁺⁴	1	2	yes	—
140	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
141	Q₈×C₆₇	—	4	16	no	—
142	D₄×C₆₇	—	4	16	no	—
143	C₂×C₄×C₆₇	—	2	8	no	—
144	C₂×C₄×C₆₇	—	2	8	no	—
145	C₂×C₄×C₆₇	—	2	8	no	—
146	C₂×C₄×C₆₇	—	2	2	no	—
147	C₂×C₄×C₆₇	—	2	4	no	—
148	C₂×C₄×C₆₇	—	2	4	no	—
149	C₂×C₄×C₆₇	—	2	2	no	—
150	C₂×C₄×C₆₇	—	2	4	no	—
151	C₂×C₄×C₆₇	—	2	4	no	—
152	C₂×C₄×C₆₇	—	2	4	no	—
153	C₂×C₄×C₆₇	—	2	4	no	—
154	C₂×C₄×C₆₇	—	2	8	no	—
155	C₂×C₄×C₆₇	—	2	8	no	—
156	C₂×C₄×C₆₇	—	2	8	no	—
157	C₂×C₄×C₆₇	—	2	8	no	—
158	C₂×C₄×C₆₇	—	2	4	no	—
159	C₂×C₄×C₆₇	—	2	8	no	—
160	C₂×C₄×C₆₇	—	4	8	no	—
161	C₂×C₄×C₆₇	—	4	8	no	—
162	C₂×C₄×C₆₇	—	4	8	no	—
163	C₂×C₄×C₆₇	—	4	16	no	—
164	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
165	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
166	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
167	C₂²×C₆₇	—	4	4	no	—
168	C₂²×C₆₇	—	4	4	no	—
169	C₂²×C₆₇	—	4	4	no	—
170	C₂²×C₆₇	—	4	8	no	—
171	C₂²×C₆₇	—	4	8	no	—
172	C₂²×C₆₇	—	8	16	no	—
173	C₂²×C₆₇	#535	1	4	yes	—
174	C₂²×C₆₇	#536	1	4	yes	—
175	C₂²×C₆₇	#537	1	4	yes	—
176	C₂²×C₆₇	#562	1	8	yes	—
177	C₂²×C₆₇	#563	1	8	yes	—
178	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
179	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
180	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
181	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	2	yes	—
182	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
183	C₄×C₆₇	—	2	4	no	—
184	C₄×C₆₇	—	2	4	no	—
185	C₄×C₆₇	—	2	4	no	—
186	C₄×C₆₇	—	4	4	no	—
187	C₄×C₆₇	—	4	4	no	—
188	C₄×C₆₇	—	4	4	no	—
189	C₄×C₆₇	—	4	8	no	—
190	C₂×#2820	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
191	C₂×C₆₇	—	8	4	no	—
192	C₂×C₆₇	#2820	1	8	yes	direct product factor
193	C₂×C₆₇	C₂×#535	1	1	yes	characteristic
194	C₂×C₆₇	C₂×#536	1	1	yes	characteristic
195	C₂×C₆₇	C₂×#537	1	1	yes	characteristic
196	C₂×C₆₇	C₂×#562	1	2	yes	—
197	C₂×C₆₇	C₂×#563	1	2	yes	—
198	C₂²×#125	C₂×C₆₇	1	1	yes	characteristic, complemented
199	C₂²×#124	C₂×C₆₇	1	1	yes	characteristic
200	#2820	C₂×C₆₇	1	16	yes	direct product factor
201	C₂×#399	C₂×C₆₇	1	2	yes	complemented
202	C₂×#397	C₂×C₆₇	1	2	yes	complemented
203	C₂×#396	C₂×C₆₇	1	4	yes	complemented
204	C₂×#395	C₂×C₆₇	1	2	yes	complemented
205	C₂×#394	C₂×C₆₇	1	2	yes	—
206	C₂×#382	C₂×C₆₇	1	1	yes	characteristic, complemented
207	C₆₇	C₂×#2820	1	1	yes	characteristic, direct product factor, Sylow
208	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
209	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
210	C₂×#125	—	2	8	no	—
211	C₂×#125	C₂²×C₆₇	1	8	yes	complemented
212	C₂×#124	—	2	8	no	—
213	C₂×#124	C₂²×C₆₇	1	8	yes	—
214	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
215	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
216	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
217	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
218	C₂²×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
219	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
220	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
221	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
222	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
223	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
224	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
225	#399	C₂²×C₆₇	1	16	yes	complemented
226	#397	C₂²×C₆₇	1	16	yes	complemented
227	#396	C₂²×C₆₇	1	32	yes	complemented
228	#395	C₂²×C₆₇	1	16	yes	complemented
229	#394	C₂²×C₆₇	1	16	yes	—
230	#382	C₂²×C₆₇	1	8	yes	complemented
231	C₂×#95	C₂²×C₆₇	1	2	yes	—
232	C₂×#95	C₂²×C₆₇	1	4	yes	—
233	C₂×#95	C₂²×C₆₇	1	2	yes	—
234	C₂⁵	C₂³×C₆₇	1	2	yes	—
235	C₂²×Q₈	—	2	4	no	—
236	C₂²×Q₈	—	4	4	no	—
237	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
238	C₂²×D₄	—	2	4	no	—
239	C₂²×D₄	—	4	4	no	—
240	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
241	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
242	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
243	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
244	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
245	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
246	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
247	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
248	#125	—	2	32	no	—
249	#124	—	2	32	no	—
250	C₂×C₄⋊C₄	—	2	8	no	—
251	C₂×C₄⋊C₄	—	2	4	no	—
252	C₂×C₄⋊C₄	—	2	4	no	—
253	C₂×C₄⋊C₄	—	2	8	no	—
254	C₂×C₄⋊C₄	—	4	4	no	—
255	C₂×C₄⋊C₄	—	4	8	no	—
256	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
257	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
258	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
259	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
260	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
261	C₂×C₄⋊C₄	C₂³×C₆₇	1	16	yes	—
262	C₂×C₂²⋊C₄	—	2	8	no	—
263	C₂×C₂²⋊C₄	—	2	8	no	—
264	C₂×C₂²⋊C₄	—	4	8	no	—
265	C₂×C₂²⋊C₄	—	4	8	no	—
266	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
267	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
268	C₂×C₂²⋊C₄	C₂³×C₆₇	1	16	yes	—
269	C₂×C₂²⋊C₄	C₂³×C₆₇	1	16	yes	—
270	C₂×C₄²	D₄×C₆₇	1	8	yes	—
271	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
272	#95	C₂³×C₆₇	1	8	yes	—
273	#95	C₂³×C₆₇	1	8	yes	—
274	#95	C₂³×C₆₇	1	16	yes	—
275	C₂⁴	—	2	4	no	—
276	C₂⁴	—	2	4	no	—
277	C₂⁴	—	2	8	no	—
278	C₂⁴	—	2	8	no	—
279	C₂⁴	—	2	4	no	—
280	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
281	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
282	C₂×Q₈	—	2	8	no	—
283	C₂×Q₈	—	4	8	no	—
284	C₂×Q₈	—	4	16	no	—
285	C₂×D₄	—	2	8	no	—
286	C₂×D₄	—	4	8	no	—
287	C₂×D₄	—	4	16	no	—
288	C₂²×C₄	—	2	2	no	—
289	C₂²×C₄	—	2	4	no	—
290	C₂²×C₄	—	2	4	no	—
291	C₂²×C₄	—	2	2	no	—
292	C₂²×C₄	—	2	4	no	—
293	C₂²×C₄	—	2	4	no	—
294	C₂²×C₄	—	2	4	no	—
295	C₂²×C₄	—	2	4	no	—
296	C₂²×C₄	—	2	8	no	—
297	C₂²×C₄	—	2	8	no	—
298	C₂²×C₄	—	2	8	no	—
299	C₂²×C₄	—	2	8	no	—
300	C₂²×C₄	—	2	4	no	—
301	C₂²×C₄	—	2	8	no	—
302	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
303	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
304	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
305	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
306	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
307	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
308	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
309	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
310	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
311	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
312	C₄⋊C₄	—	2	16	no	—
313	C₄⋊C₄	—	2	8	no	—
314	C₄⋊C₄	—	2	8	no	—
315	C₄⋊C₄	—	2	16	no	—
316	C₄⋊C₄	—	4	8	no	—
317	C₄⋊C₄	—	4	16	no	—
318	C₂²⋊C₄	—	2	16	no	—
319	C₂²⋊C₄	—	2	16	no	—
320	C₂²⋊C₄	—	4	16	no	—
321	C₂²⋊C₄	—	4	16	no	—
322	C₄²	C₂×D₄×C₆₇	1	16	yes	—
323	C₂³	—	2	4	no	—
324	C₂³	—	2	4	no	—
325	C₂³	—	2	8	no	—
326	C₂³	—	2	8	no	—
327	C₂³	—	2	4	no	—
328	C₂³	—	4	8	no	—
329	C₂³	—	4	8	no	—
330	C₂³	—	4	8	no	—
331	C₂³	—	4	16	no	—
332	C₂³	—	4	16	no	—
333	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
334	C₂³	H(5,𝔽₂)×C₆₇	1	4	yes	—
335	C₂³	H(5,𝔽₂)×C₆₇	1	2	yes	—
336	C₂³	2₋¹⁺⁴×C₆₇	1	4	yes	—
337	C₂³	2₋¹⁺⁴×C₆₇	1	2	yes	—
338	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
339	Q₈	—	4	16	no	—
340	D₄	—	4	16	no	—
341	C₂×C₄	—	2	8	no	—
342	C₂×C₄	—	2	8	no	—
343	C₂×C₄	—	2	8	no	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	2	4	no	—
346	C₂×C₄	—	2	4	no	—
347	C₂×C₄	—	2	2	no	—
348	C₂×C₄	—	2	4	no	—
349	C₂×C₄	—	2	4	no	—
350	C₂×C₄	—	2	4	no	—
351	C₂×C₄	—	2	4	no	—
352	C₂×C₄	—	2	8	no	—
353	C₂×C₄	—	2	8	no	—
354	C₂×C₄	—	2	8	no	—
355	C₂×C₄	—	2	8	no	—
356	C₂×C₄	—	2	8	no	—
357	C₂×C₄	—	2	4	no	—
358	C₂×C₄	—	4	8	no	—
359	C₂×C₄	—	4	8	no	—
360	C₂×C₄	—	4	8	no	—
361	C₂×C₄	—	4	16	no	—
362	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
363	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
364	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
365	C₂²	—	4	8	no	—
366	C₂²	—	4	8	no	—
367	C₂²	—	4	4	no	—
368	C₂²	—	4	4	no	—
369	C₂²	—	4	4	no	—
370	C₂²	—	8	16	no	—
371	C₂²	#535×C₆₇	1	4	yes	—
372	C₂²	#536×C₆₇	1	4	yes	—
373	C₂²	#537×C₆₇	1	4	yes	—
374	C₂²	#562×C₆₇	1	8	yes	—
375	C₂²	#563×C₆₇	1	8	yes	—
376	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
377	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
378	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	2	yes	—
379	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	2	yes	—
380	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
381	C₄	—	2	4	no	—
382	C₄	—	2	4	no	—
383	C₄	—	2	4	no	—
384	C₄	—	4	8	no	—
385	C₄	—	4	4	no	—
386	C₄	—	4	4	no	—
387	C₄	—	4	4	no	—
388	C₂	—	8	4	no	—
389	C₂	C₆₇×#2820	1	8	yes	direct product factor
390	C₂	C₂×#535×C₆₇	1	1	yes	characteristic
391	C₂	C₂×#536×C₆₇	1	1	yes	characteristic
392	C₂	C₂×#537×C₆₇	1	1	yes	characteristic
393	C₂	C₂×#562×C₆₇	1	2	yes	—
394	C₂	C₂×#563×C₆₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₆₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
prime radical	C₂³×D₄×C₆₇	C₂²
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
1. C₂²×#125×C₆₇	C₂	semidirect
3. C₆₇×#2820	C₂	direct (N × H)
4. C₂×#399×C₆₇	C₂	semidirect
5. C₂×#397×C₆₇	C₂	semidirect
6. C₂×#396×C₆₇	C₂	semidirect
7. C₂×#395×C₆₇	C₂	semidirect
9. C₂×#382×C₆₇	C₂	semidirect
13. C₂×#125×C₆₇	C₂²	semidirect
27. #399×C₆₇	C₂²	semidirect
28. #397×C₆₇	C₂²	semidirect
29. #396×C₆₇	C₂²	semidirect
30. #395×C₆₇	C₂²	semidirect
32. #382×C₆₇	C₂²	semidirect
190. C₂×#2820	C₆₇	direct (N × H)
192. C₂×C₆₇	#2820	direct (N × H)
198. C₂²×#125	C₂×C₆₇	semidirect
200. #2820	C₂×C₆₇	direct (N × H)
201. C₂×#399	C₂×C₆₇	semidirect
202. C₂×#397	C₂×C₆₇	semidirect
203. C₂×#396	C₂×C₆₇	semidirect
204. C₂×#395	C₂×C₆₇	semidirect
206. C₂×#382	C₂×C₆₇	semidirect
207. C₆₇	C₂×#2820	direct (N × H)
211. C₂×#125	C₂²×C₆₇	semidirect
225. #399	C₂²×C₆₇	semidirect
226. #397	C₂²×C₆₇	semidirect
227. #396	C₂²×C₆₇	semidirect
228. #395	C₂²×C₆₇	semidirect
230. #382	C₂²×C₆₇	semidirect
389. C₂	C₆₇×#2820	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×#125×C₆₇	2
C₂²×#124×C₆₇	2
C₆₇×#2820	2
C₂×#399×C₆₇	2
C₂×#397×C₆₇	2
C₂×#396×C₆₇	2
C₂×#395×C₆₇	2
C₂×#394×C₆₇	2
C₂×#382×C₆₇	2
C₂×#2820	67

Subgroups — C2×C67×#2820 #30715932

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2820 #30715932